GCSE Maths Volume, Surface Area and 3D Shapes (Foundation): worked examples and questions with answers
By the end of this chapter you will be able to
- Name 3D solids and count their faces, edges and vertices
- Recognise and draw nets of cuboids, prisms, pyramids and cylinders
- Draw and use plans and front and side elevations
- Work out the volume of cuboids, prisms and cylinders, and use units of volume and capacity
- Use density = mass ÷ volume
- Work out the surface area of cuboids, prisms and cylinders
You should already know
- Area of rectangles, triangles, trapezia and circles (Chapter 18)
- Metric units of length and capacity
- Powers and square roots (Chapters 1 and 2)
Where this leads
- Pythagoras and trigonometry in 3D (Chapter 20)
- Similar shapes and scale factors for area and volume (Chapter 21)
- Density, pressure and compound measures (Chapter 16)
Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references: G12, G13, G16, G17.
- 19.1 Properties of 3D solids: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.2 Nets of solids: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.3 Plans and elevations: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.4 Volume of cuboids, and units of volume and capacity: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.5 Volume of prisms: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.6 Volume of cylinders, and density: Worksheet (PDF) · Answers (PDF) · lesson page
- 19.7 Surface area of cuboids, prisms and cylinders: Worksheet (PDF) · Answers (PDF) · lesson page
- Check-up and exam-style questions: Worksheet (PDF) · Answers (PDF) · lesson page
Exercise P Do you remember?
Non-calculatorMethod. Area of a rectangle = length × width.
- Write the formula and put in the numbers. Area is the space inside a flat shape.Area = 9 × 4
- Multiply. 9 × 4 = 36
- Write the unit. Area uses square units.Area = 36 cm2
Check. 4 rows of 9 squares: 9 + 9 + 9 + 9 = 36. ✓
Watch out. Area is in square units (cm2).
Method. Area of a triangle = ½ × base × perpendicular height.
- Write the formula and substitute. A triangle is half of a rectangle with the same base and height.Area = ½ × 10 × 6
- Multiply 10 by 6, then halve. 10 × 6 = 60, and 60 ÷ 2 = 30
- Write the unit. Area = 30 cm2
Check. The rectangle 10 × 6 has area 60, and half of 60 is 30. ✓
Watch out. Use the height at right angles to the base, not a sloping side.
Method. Area of a circle = πr2.
- Write the formula and substitute r = 5. r2 means r × r.Area = π × 52
- Square the radius. 52 = 25
- Write it in terms of π. Keep the π symbol instead of multiplying by 3.14.Area = 25π cm2
Check. 25π ≈ 25 × 3.14 = 78.5, which is a sensible area for a circle that is 10 cm across. ✓
Watch out. Square the radius only: πr2 is not (πr)2.
Method. Circumference = π × diameter.
- Write the formula. The circumference is the distance round the circle.C = πd
- Substitute d = 12. C = π × 12
- Write it in terms of π. Put the number in front of the π.C = 12π cm
Check. Using 2πr with r = 6: 2 × 6 = 12, so 12π. ✓
Watch out. Circumference is a length (cm), area is cm2.
Method. 1 litre = 1000 millilitres, so multiply by 1000.
- State the fact. There are 1000 ml in every litre.1 litre = 1000 ml
- Multiply. Changing to a smaller unit gives a bigger number.3.5 × 1000 = 3500
- Write the unit. 3500 ml
Check. 3500 ÷ 1000 = 3.5 litres. ✓
Watch out. Do not multiply by 100: litres use 1000.
Properties of 3D solids
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A flat shape such as a rectangle or a circle is 2D: it has length and width only. A 3D shape, also called a solid, takes up space, so it also has depth or height. A cereal box, a can of beans, a football and a tent are all solids. In Chapter 18 you worked with the flat shapes; in this chapter you meet the solids built from them.
To describe a solid we use three words. A face is a flat surface (the top of a box is a face). An edge is a line where two faces meet. A vertex is a corner where edges meet; the plural is vertices. Some solids also have a curved surface, which is not a flat face: a tin has one curved surface round its side.
Two families of solid are very important. A prism has the same shape (its cross-section) all the way along its length. A cuboid is a prism with a rectangle as its cross-section; a triangular prism has a triangle. A pyramid has a flat base and triangular faces that rise to a single point called the apex. We name them by the shape of the base: a square-based pyramid has a square base, a hexagonal pyramid has a hexagon.
Three solids have curved surfaces. A cylinder has two flat circular faces and one curved surface. A cone has one flat circular face, one curved surface and one vertex (its point). A sphere is a perfect ball: one curved surface and no edges or vertices.
Counting faces, edges and vertices gives a pattern. Look at the last column of this table:
| Solid | Faces (F) | Edges (E) | Vertices (V) | V − E + F |
|---|---|---|---|---|
| Cuboid or cube (square or rectangular prism) | 6 | 12 | 8 | 8 − 12 + 6 = 2 |
| Triangular prism | 5 | 9 | 6 | 6 − 9 + 5 = 2 |
| Pentagonal prism | 7 | 15 | 10 | 10 − 15 + 7 = 2 |
| Hexagonal prism | 8 | 18 | 12 | 12 − 18 + 8 = 2 |
| Triangular-based pyramid (tetrahedron) | 4 | 6 | 4 | 4 − 6 + 4 = 2 |
| Square-based pyramid | 5 | 8 | 5 | 5 − 8 + 5 = 2 |
| Pentagonal-based pyramid | 6 | 10 | 6 | 6 − 10 + 6 = 2 |
| Hexagonal-based pyramid | 7 | 12 | 7 | 7 − 12 + 7 = 2 |
For every solid with flat faces and no holes, vertices − edges + faces = 2. This is called Euler’s formula (say “Oiler”). It is useful as a check on your counting, and for finding a missing number.
Key ideas
- Face: a flat surface. Edge: where two faces meet. Vertex: a corner.
- A prism has a constant cross-section; a pyramid has a base and triangular faces meeting at the apex.
- Cylinder: 2 flat faces, 1 curved surface. Cone: 1 flat face, 1 curved surface, 1 vertex. Sphere: 1 curved surface only.
- An n-sided prism has n + 2 faces, 3n edges and 2n vertices. An n-sided pyramid has n + 1 faces, 2n edges and n + 1 vertices.
- Euler’s formula: V − E + F = 2.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a triangular prism.
- Count the faces. A prism has 2 identical end faces and one rectangular face for each side of an end. The ends have 3 sides.2 ends + 3 rectangles = 5 faces
- Count the edges. Each end has 3 edges (that is 3 + 3) and 3 more edges join the two ends together.3 + 3 + 3 = 9 edges
- Count the vertices. Each end has 3 corners and there are two ends.3 + 3 = 6 vertices
Check. Euler’s formula: vertices − edges + faces = 6 − 9 + 5 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Method. Use the faces to identify the family of solid, then count the corners.
- Look at the faces. One square face and four triangles that meet at a point means a pyramid with a square base.Solid: square-based pyramid
- Count the vertices. There are 4 corners on the square base and 1 more at the top (the apex).4 + 1 = 5
- State the answer. A square-based pyramid has 5 vertices.
Check. Euler’s formula: 5 − 8 + 5 = 2 (a square-based pyramid has 8 edges). ✓
Watch out. Do not count the apex twice, and do not forget it.
Method. Use Euler’s formula: V − E + F = 2, and solve for the missing number.
- Write Euler’s formula. For a solid with flat faces and no holes: vertices − edges + faces = 2.V − E + F = 2
- Put in the numbers you know. V − 16 + 9 = 2
- Simplify the known part. V − 7 = 2
- Solve for V. Add the same to both sides.V = 2 + 7 = 9
Check. 9 − 16 + 9 = 2. ✓
Watch out. Be careful with the minus sign in front of E.
Watch out
Count the faces of a prism by splitting it up: the two ends plus the sides. A common slip is to forget the ends, or to count the same edge twice. A cylinder, cone and sphere have curved surfaces, so they do not have faces and edges in the same way as a cube.Exercise 19A Properties of 3D solids
Non-calculatorMethod. Look at the shape of the solid and match it to the family it belongs to.
- Look for the clue. It has triangular ends and rectangular faces, with the same triangle all along its length.
- Name it. The solid is a triangular prism.
Check. Compare with the pictures at the start of this section. ✓
Method. Look at the shape of the solid and match it to the family it belongs to.
- Look for the clue. It is perfectly round with no flat faces.
- Name it. The solid is a sphere.
Check. Compare with the pictures at the start of this section. ✓
Method. Look at the shape of the solid and match it to the family it belongs to.
- Look for the clue. It has one flat circular base and a curved surface that comes to a point.
- Name it. The solid is a cone.
Check. Compare with the pictures at the start of this section. ✓
Method. Look at the shape of the solid and match it to the family it belongs to.
- Look for the clue. All six faces are rectangles.
- Name it. The solid is a cuboid.
Check. Compare with the pictures at the start of this section. ✓
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a square prism (a cuboid or cube).
- Count the faces. A prism has 2 identical end faces and one rectangular face for each side of an end. The ends have 4 sides.2 ends + 4 rectangles = 6 faces
- Count the edges. Each end has 4 edges (that is 4 + 4) and 4 more edges join the two ends together.4 + 4 + 4 = 12 edges
- Count the vertices. Each end has 4 corners and there are two ends.4 + 4 = 8 vertices
Check. Euler’s formula: vertices − edges + faces = 8 − 12 + 6 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a triangular prism.
- Count the faces. A prism has 2 identical end faces and one rectangular face for each side of an end. The ends have 3 sides.2 ends + 3 rectangles = 5 faces
- Count the edges. Each end has 3 edges (that is 3 + 3) and 3 more edges join the two ends together.3 + 3 + 3 = 9 edges
- Count the vertices. Each end has 3 corners and there are two ends.3 + 3 = 6 vertices
Check. Euler’s formula: vertices − edges + faces = 6 − 9 + 5 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a square pyramid.
- Count the faces. A pyramid has 1 base with 4 sides, and one triangular face on each side of the base.1 base + 4 triangles = 5 faces
- Count the edges. The base has 4 edges. Another 4 sloping edges go from the base corners up to the apex.4 + 4 = 8 edges
- Count the vertices. There is a vertex at each corner of the base (4) and one more at the apex.4 + 1 = 5 vertices
Check. Euler’s formula: vertices − edges + faces = 5 − 8 + 5 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a hexagonal prism.
- Count the faces. A prism has 2 identical end faces and one rectangular face for each side of an end. The ends have 6 sides.2 ends + 6 rectangles = 8 faces
- Count the edges. Each end has 6 edges (that is 6 + 6) and 6 more edges join the two ends together.6 + 6 + 6 = 18 edges
- Count the vertices. Each end has 6 corners and there are two ends.6 + 6 = 12 vertices
Check. Euler’s formula: vertices − edges + faces = 12 − 18 + 8 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Method. Look at the cylinder and think about which parts are flat, which are curved, and where surfaces meet.
- Count the flat faces and curved surfaces. It has a flat circular face at the top and another at the bottom, and one curved surface that wraps round the side.Flat faces: 2. Curved surfaces: 1.
- Count the edges and the vertices. Each flat circular face meets the curved surface along a circular edge: one edge at the top and one at the bottom. It has no corners.Edges: 2. Vertices: 0.
Check. A solid like this has only 3 surface(s) in total, so it is a good idea to say “flat” or “curved” when you describe them. ✓
Watch out. A curved surface is not a flat face, but it still counts as a surface.
Method. Look at the cone and think about which parts are flat, which are curved, and where surfaces meet.
- Count the flat faces and curved surfaces. It has one flat circular face (the base) and one curved surface that rises to a point.Flat faces: 1. Curved surfaces: 1.
- Count the edges and the vertices. The base meets the curved surface along one circular edge. The pointed top is the only vertex.Edges: 1. Vertices: 1.
Check. A solid like this has only 2 surface(s) in total, so it is a good idea to say “flat” or “curved” when you describe them. ✓
Watch out. A curved surface is not a flat face, but it still counts as a surface.
Method. Look at the sphere and think about which parts are flat, which are curved, and where surfaces meet.
- Count the flat faces and curved surfaces. It has no flat faces at all: it is one smooth curved surface.Flat faces: 0. Curved surfaces: 1.
- Count the edges and the vertices. There is no line where two surfaces meet and there are no corners, so no edges and no vertices.Edges: 0. Vertices: 0.
Check. A solid like this has only 1 surface(s) in total, so it is a good idea to say “flat” or “curved” when you describe them. ✓
Watch out. A curved surface is not a flat face, but it still counts as a surface.
Method. A solid made only of squares needs to have equal edges.
- Match the clue. Six faces that are all squares of the same size make a cube.
- Name it. The solid is a cube.
Check. A cube has 6 faces, 12 edges and 8 vertices. ✓
Method. Two identical ends and rectangles on the sides describe a prism.
- Match the clue. The ends are triangles, so it is a triangular prism. Its three sides are rectangles.
- Name it. The solid is a triangular prism.
Check. 2 + 3 = 5 faces, which matches the table. ✓
Method. A base and triangular faces meeting at a point describe a pyramid.
- Match the clue. The base is a square, so the pyramid is square-based.
- Name it. The solid is a square-based pyramid.
Check. 1 + 4 = 5 faces. ✓
Method. A curved surface and one point at the top describe a cone.
- Match the clue. The only solid with one circular base and one vertex is a cone.
- Name it. The solid is a cone.
Check. A cylinder has two circular faces and no vertex, so it is not a cylinder. ✓
Method. Add up the lengths of all the edges. Matching edges come in groups of four.
- Find the total of one edge from each group. There are three different lengths: 5, 3 and 2.5 + 3 + 2 = 10 cm
- Multiply by four. Each length appears four times, so the total is 4 lots of this sum.4 × 10 = 40 cm
- Write the answer. Total length = 40 cm
Check. Add each group separately: 4 × 5 = 20, 4 × 3 = 12, 4 × 2 = 8, and 20 + 12 + 8 = 40. ✓
Watch out. A cuboid has 12 edges, not 6 or 8.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a pentagonal prism.
- Count the faces. A prism has 2 identical end faces and one rectangular face for each side of an end. The ends have 5 sides.2 ends + 5 rectangles = 7 faces
- Count the edges. Each end has 5 edges (that is 5 + 5) and 5 more edges join the two ends together.5 + 5 + 5 = 15 edges
- Count the vertices. Each end has 5 corners and there are two ends.5 + 5 = 10 vertices
Check. Euler’s formula: vertices − edges + faces = 10 − 15 + 7 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a pentagonal pyramid.
- Count the faces. A pyramid has 1 base with 5 sides, and one triangular face on each side of the base.1 base + 5 triangles = 6 faces
- Count the edges. The base has 5 edges. Another 5 sloping edges go from the base corners up to the apex.5 + 5 = 10 edges
- Count the vertices. There is a vertex at each corner of the base (5) and one more at the apex.5 + 1 = 6 vertices
Check. Euler’s formula: vertices − edges + faces = 6 − 10 + 6 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Key words. A face is a flat surface. An edge is where two faces meet. A vertex (plural vertices) is a corner. A prism has the same cross-section all along its length. A pyramid has a base and triangular faces meeting at a point (the apex).
Method. Count faces, edges and vertices in turn. Split the solid into its base(s) and sides to count without missing any.
- Name the solid. It helps to picture or sketch it.The solid is a triangular-based pyramid (a tetrahedron).
- Count the faces. A pyramid has 1 base with 3 sides, and one triangular face on each side of the base.1 base + 3 triangles = 4 faces
- Count the edges. The base has 3 edges. Another 3 sloping edges go from the base corners up to the apex.3 + 3 = 6 edges
- Count the vertices. There is a vertex at each corner of the base (3) and one more at the apex.3 + 1 = 4 vertices
Check. Euler’s formula: vertices − edges + faces = 4 − 6 + 4 = 2. ✓
Watch out. Count each edge once only, and remember the base (or both ends) when you count faces.
Method. A square-based pyramid has 4 edges on the base and 4 sloping edges.
- Count the straws of each kind. The base is a square with 4 sides. 4 more edges go up to the apex.4 base edges and 4 sloping edges (8 edges in total)
- Find the total length of the base straws. 4 × 8 = 32 cm
- Find the total length of the sloping straws. 4 × 9 = 36 cm
- Add the two totals. 32 + 36 = 68 cm
Check. Using 8 edges: 4 × (8 + 9) = 4 × 17 = 68. ✓
Watch out. Do not forget the sloping edges or count only 5 straws.
Method. Compare the numbers of faces, edges and vertices with the table at the start of this section.
- Write the counts. Faces = 5, edges = 9, vertices = 6
- Find the matching solid in the table. The numbers of faces, edges and vertices are different for each solid, so only one fits.5, 9, 6 matches the triangular prism.
Check. Euler: 6 − 9 + 5 = 2. ✓
Method. Compare the numbers of faces, edges and vertices with the table at the start of this section.
- Write the counts. Faces = 5, edges = 8, vertices = 5
- Find the matching solid in the table. The numbers of faces, edges and vertices are different for each solid, so only one fits.5, 8, 5 matches the square-based pyramid.
Check. Euler: 5 − 8 + 5 = 2. ✓
Method. Compare the numbers of faces, edges and vertices with the table at the start of this section.
- Write the counts. Faces = 4, edges = 6, vertices = 4
- Find the matching solid in the table. The numbers of faces, edges and vertices are different for each solid, so only one fits.4, 6, 4 matches the triangular-based pyramid (tetrahedron).
Check. Euler: 4 − 6 + 4 = 2. ✓
Method. Compare the numbers of faces, edges and vertices with the table at the start of this section.
- Write the counts. Faces = 7, edges = 15, vertices = 10
- Find the matching solid in the table. The numbers of faces, edges and vertices are different for each solid, so only one fits.7, 15, 10 matches the pentagonal prism.
Check. Euler: 10 − 15 + 7 = 2. ✓
Method. Use Euler’s formula: V − E + F = 2, and solve for the missing number.
- Write Euler’s formula. For a solid with flat faces and no holes: vertices − edges + faces = 2.V − E + F = 2
- Put in the numbers you know. V − 18 + 8 = 2
- Simplify the known part. V − 10 = 2
- Solve for V. Add the same to both sides.V = 2 + 10 = 12
Check. 12 − 18 + 8 = 2. ✓
Watch out. Be careful with the minus sign in front of E.
Method. Use Euler’s formula: V − E + F = 2, and solve for the missing number.
- Write Euler’s formula. For a solid with flat faces and no holes: vertices − edges + faces = 2.V − E + F = 2
- Put in the numbers you know. 12 − E + 8 = 2
- Combine the known numbers. 20 − E = 2
- Solve for E. Rearrange: E is the total minus 2.E = 20 − 2 = 18
Check. 12 − 18 + 8 = 2. ✓
Watch out. Be careful with the minus sign in front of E.
Method. Use Euler’s formula: V − E + F = 2, and solve for the missing number.
- Write Euler’s formula. For a solid with flat faces and no holes: vertices − edges + faces = 2.V − E + F = 2
- Put in the numbers you know. 5 − 8 + F = 2
- Combine the known numbers. −3 + F = 2
- Solve for F. Add or subtract the same to both sides.F = 2 + 3 = 5
Check. 5 − 8 + 5 = 2. ✓
Watch out. Be careful with the minus sign in front of E.
Method. A prism whose ends have n sides has 3n edges, so solve 3n = 24.
- Write the equation. The edges are n on each end plus n joining the ends.3n = 24
- Divide by 3. Undo the multiplication.n = 24 ÷ 3 = 8
Check. 3 × 8 = 24. ✓ (an octagonal prism)
Method. A prism has n + 2 faces: two ends and n rectangles.
- Use n = 8. From part (a) the ends are octagons with 8 sides.n = 8
- Add 2 for the ends. 8 + 2 = 10 faces
Check. Euler: V = 16, E = 24, F = 10 gives 16 − 24 + 10 = 2. ✓
Method. A pyramid has 1 base and one triangular face for each side of the base, so faces = n + 1 where n is the number of sides of the base.
- Find the number of sides of the base. 10 faces = 1 base + n triangles.n = 10 − 1 = 9 (a nine-sided base)
- Find the edges. The base has 9 edges and 9 more edges go up to the apex.9 + 9 = 18 edges
- Find the vertices. There are 9 corners on the base and 1 at the apex.9 + 1 = 10 vertices
Check. Euler: V − E + F = 10 − 18 + 10 = 2. ✓
Watch out. The base counts as one of the faces.
Method. Describe the surfaces of the cone, then compare with Ali’s numbers.
- List the surfaces. A cone has one flat circular base and one curved surface.1 flat face + 1 curved surface = 2 surfaces
- List the edge and the vertex. The base meets the curved surface in one circular edge, and the point at the top is the only vertex.1 edge and 1 vertex
- Compare with Ali. Ali is right if the curved surface is counted as a face. Many textbooks say “1 flat face and 1 curved surface” to be clear.Ali is correct.
Check. Edexcel may ask for “flat faces” or “curved surfaces”: use those words to avoid confusion. ✓
Watch out. Say “flat face” when you mean a flat surface.
Method. Test the statement with an example from the table.
- Think of an example. All the faces of a cube are squares, and a square is a special rectangle, so it is a cuboid with equal edges.
- Decide. The statement is True.
Check. Checking with a sketch of the solid agrees. ✓
Method. Test the statement with an example from the table.
- Think of an example. A sphere has one smooth curved surface, so no two surfaces meet in an edge, and there are no corners.
- Decide. The statement is True.
Check. Checking with a sketch of the solid agrees. ✓
Method. Test the statement with an example from the table.
- Think of an example. A triangular prism has a triangle as its cross-section, for example, so this is not always true.
- Decide. The statement is False.
Check. Checking with a sketch of the solid agrees. ✓
Method. Test the statement with an example from the table.
- Think of an example. A square-based pyramid has 5 faces (the base and 4 triangles), and 5 is odd.
- Decide. The statement is False.
Check. Checking with a sketch of the solid agrees. ✓
Method. Use Euler’s formula: V − E + F = 2, and solve for the missing number.
- Write Euler’s formula. For a solid with flat faces and no holes: vertices − edges + faces = 2.V − E + F = 2
- Put in the numbers you know. 6 − E + 10 = 2
- Combine the known numbers. 16 − E = 2
- Solve for E. Rearrange: E is the total minus 2.E = 16 − 2 = 14
Check. 6 − 14 + 10 = 2. ✓
Watch out. Be careful with the minus sign in front of E.
Method. A prism with an n-sided end has 2n vertices, 3n edges and n + 2 faces. Test these against the numbers.
- Find n from the vertices. 2n = 6, so n = 3.n = 3
- Work out the faces of a triangular prism. n + 2 = 3 + 2 = 5 faces, but the solid has 10 faces.5 ≠ 10
- Conclude. The solid cannot be a prism.
Check. A prism with 10 faces would have 8-sided ends and 16 vertices, not 6. ✓
Method. Think about the shape of the tin.
- Look at the shape. A tin has two flat circles and a curved side.
- Name it. cylinder
Check. The same shape as a drinks can. ✓
Method. Think about the shape of the ball.
- Look at the shape. A football is perfectly round.
- Name it. sphere
Check. No flat faces and no edges. ✓
Method. Think about the shape of the dice.
- Look at the shape. All six faces are equal squares.
- Name it. cube
Check. 6 faces, 12 edges and 8 vertices. ✓
Method. Think about the shape of the cone.
- Look at the shape. It has a circular opening and comes to a point.
- Name it. cone
Check. One circular face and one vertex. ✓
Method. Picture a dice and count each part.
- Faces. A dice has a face on top, on the bottom and on each of the four sides.1 + 1 + 4 = 6
- Edges. 4 round the top, 4 round the bottom, and 4 going up the sides.4 + 4 + 4 = 12
- Vertices. 4 corners on top and 4 corners on the bottom.4 + 4 = 8
Check. Euler: 8 − 12 + 6 = 2. ✓
Method. Go through the solids and decide if any surface is not flat.
- Check the solids with only flat faces. A cube, a triangular prism and a square-based pyramid are made only of squares, rectangles and triangles.Not curved: cube, triangular prism, square-based pyramid
- Check the others. A cylinder has a curved side. A sphere is curved all over.Curved: cylinder and sphere
Check. A cone is also curved, but it was not in the list. ✓
Nets of solids
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A net is a flat pattern that folds up to make a solid. Imagine cutting along some edges of a cardboard box and unfolding it flat: the flat shape you get is a net. Nets show you every face of a solid at once, which makes them very useful for finding surface areas (Section 19.7).
Every net is made from the faces of the solid. A cuboid has six rectangular faces, so its net has six rectangles. A triangular prism has two triangles and three rectangles. A square-based pyramid has one square and four triangles. A cylinder has two circles and one rectangle. Counting the faces helps you decide which solid a net makes.
Nets of a cube. A cube has 6 identical square faces, so its net is 6 squares joined edge to edge. But not every arrangement of 6 squares folds into a cube. There are exactly 11 different nets of a cube. To test a net, imagine folding it: if two squares land on the same face, it fails. Squares in a straight row of four wrap round the cube like a belt, so a row of 5 or a 2 × 3 block never works.
Nets of cylinders. If you peel the label off a tin of soup and lay it flat, you get a rectangle. Its height is the height of the tin. Its width is the distance once round the tin, which is the circumference of the circular end, 2πr. So a cylinder’s net is a rectangle (2πr by h) with a circle of radius r touching the top edge and another touching the bottom edge.
Key ideas
- A net is a flat pattern that folds to make a solid; it shows every face.
- There are 11 different nets of a cube. If two squares fold onto the same face, the net fails.
- Cuboid: 6 rectangles. Triangular prism: 2 triangles + 3 rectangles. Square-based pyramid: 1 square + 4 triangles. Cylinder: 2 circles + 1 rectangle.
- The rectangle in the net of a cylinder has width 2πr (the circumference) and height h.
- Higher: the sector in the net of a cone has radius l and angle 360 × r ÷ l.
Method. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite D; B is opposite E; C is opposite F
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.5 × 3 = 15 (and there are 2 of them, so 30)
- Front and back. Each is length × height.5 × 2 = 10 (2 of them: 20)
- Left and right. Each is width × height.3 × 2 = 6 (2 of them: 12)
- Add the three pairs. The surface area is every face added together.30 + 20 + 12 = 62
- Write the unit. Area, so square units.Surface area = 62 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (15 + 10 + 6) = 2 × 31 = 62. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Method. The rectangle in the net of a cylinder is as wide as the circumference of the circular end.
- Think about how the rectangle wraps round. When the net is folded up, the long edge of the rectangle goes right round the circular end once. So it is as long as the circumference of the circle.
- Use the circumference formula. Circumference = 2πr (or πd).C = 2 × π × 3 = 6π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Width = 6π cm
Check. Circumference is a bit more than 3 × the diameter: 3 × 6 = 18. ✓
Watch out. Use the circumference (2πr) for the width of the rectangle, not the area of the circle.
Watch out
Squares that touch in a net do not always touch on the cube, and squares that look far apart can be opposite. Always fold the net in your head one square at a time. For a cylinder the width of the rectangle is the circumference (2πr), not the radius and not the area of the circle.Exercise 19B Nets of solids
Non-calculatorMethod. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite F; C is opposite E; D is opposite B
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Method. Mentally fold the net up. If two squares land on the same face, the net does not make a cube.
- Pick one square to be the bottom of the cube. Imagine holding it flat on a table and folding the other squares up around it.
- Fold the other squares round, one at a time. Each square becomes a face of the cube. We follow where each one lands.
- Look for squares that land on top of each other. A cube has 6 different faces. If two squares fold onto the same face, one face is covered twice and another face is left open.Squares C and D fold onto the same face, so there is a hole in the cube.
Check. Count the faces round the net: a cube needs each of its 6 faces exactly once. ✓
Watch out. A net needs more than six squares joined together: they must also fold into six different faces.
Method. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite D; B is opposite E; C is opposite F
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Method. Mentally fold the net up. If two squares land on the same face, the net does not make a cube.
- Pick one square to be the bottom of the cube. Imagine holding it flat on a table and folding the other squares up around it.
- Fold the other squares round, one at a time. Each square becomes a face of the cube. We follow where each one lands.
- Look for squares that land on top of each other. A cube has 6 different faces. If two squares fold onto the same face, one face is covered twice and another face is left open.Squares A and E fold onto the same face, so there is a hole in the cube.
Check. Count the faces round the net: a cube needs each of its 6 faces exactly once. ✓
Watch out. A net needs more than six squares joined together: they must also fold into six different faces.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face A. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite A is F.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face B. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite B is D.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face F. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite F is A.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has six rectangles.
- Name the solid. The net makes a cuboid.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has two triangles and three rectangles.
- Name the solid. The net makes a triangular prism.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has one square and four triangles.
- Name the solid. The net makes a square-based pyramid.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has two circles and one rectangle.
- Name the solid. The net makes a cylinder.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. A prism has two end faces and a rectangle for each side of the end.
- Count the ends. The ends are triangles.2 triangles
- Count the rectangles. A triangle has 3 sides, so there are 3 rectangles.3 rectangles
Check. 2 + 3 = 5 faces. ✓
Method. A pyramid has one base and a triangle on each side of the base.
- Count the base. The base is a square.1 square
- Count the triangles. A square has 4 sides.4 triangles
Check. 1 + 4 = 5 faces. ✓
Method. A prism has two end faces and a rectangle for each side of the end.
- Count the ends. The ends are pentagons.2 pentagons
- Count the rectangles. A pentagon has 5 sides.5 rectangles
Check. 2 + 5 = 7 faces; Euler gives 10 − 15 + 7 = 2. ✓
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.6 × 4 = 24 (and there are 2 of them, so 48)
- Front and back. Each is length × height.6 × 3 = 18 (2 of them: 36)
- Left and right. Each is width × height.4 × 3 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.48 + 36 + 24 = 108
- Write the unit. Area, so square units.Surface area = 108 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (24 + 18 + 12) = 2 × 54 = 108. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.
- Area of one face. Each face is a square with side equal to the edge length.5 × 5 = 25
- Multiply by the number of faces. A cube has 6 faces.6 × 25 = 150
- Write the unit. Surface area = 150 cm2
Check. Formula 6a2 = 6 × 52 = 6 × 25 = 150. ✓
Watch out. Do not confuse with the volume: volume cubes the side (a3), surface area squares it (6a2).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. Use the perpendicular height of the triangle, not the sloping side.½ × 6 × 4 = 12 cm2
- Double it for the two ends. The two end faces are identical.2 × 12 = 24 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle under the base: 6 × 10 = 60
First sloping rectangle: 5 × 10 = 50
Second sloping rectangle: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.24 + 60 + 50 + 50 = 184
- Write the unit. Area, so square units.Surface area = 184 cm2
Check. The rectangles together make (perimeter of an end) × length = 16 × 10 = 160. Then 24 + 160 = 184. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Method. The rectangle in the net of a cylinder is as wide as the circumference of the circular end.
- Think about how the rectangle wraps round. When the net is folded up, the long edge of the rectangle goes right round the circular end once. So it is as long as the circumference of the circle.
- Use the circumference formula. Circumference = 2πr (or πd).C = 2 × π × 4 = 8π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Width = 8π cm
Check. Circumference is a bit more than 3 × the diameter: 3 × 8 = 24. ✓
Watch out. Use the circumference (2πr) for the width of the rectangle, not the area of the circle.
Method. Replace π in 8π by 3.14 and multiply.
- Write the exact width from part (a). The width is the circumference.8π cm
- Replace π by 3.14. This gives an estimate of the width.8 × 3.14 = 25.12
Check. 8 × 3 = 24, and 25.12 is just over 24. ✓
Method. The label wraps once round the tin, so its length is the circumference.
- Write the circumference formula. C = 2πr.C = 2 × 22/7 × 3.5
- Work it out. 2 × 3.5 = 7, and 7 × 22/7 = 22.C = 7 × 22/7 = 22 cm
Check. The diameter is 7 cm and 7 × 22/7 = 22. ✓
Method. The label is a rectangle: length (the circumference) × height of the tin.
- Write the sizes. The length is 22 cm from part (a) and the height is 10 cm.22 cm by 10 cm
- Multiply. Area of a rectangle = length × width.22 × 10 = 220 cm2
Check. Area of curved surface = 2πrh = 2 × 22/7 × 3.5 × 10 = 220. ✓
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 8 × 5 × 3
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.8 × 5 = 40
- Multiply by the height. This counts all the layers.40 × 3 = 120
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 120 cm3
Check. Multiply in a different order: 5 × 3 = 15, then 15 × 8 = 120. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Method. Fold the net up in your head, starting from one end of the row.
- Fold the row of five. Four squares in a row wrap round to make the four side faces of the cube, like a belt.Squares 1, 2, 3 and 4 make the sides.
- Fold in the fifth square. It keeps going round the same belt, so it lands on top of square 1.Square 5 lands on square 1.
- State the conclusion. Two squares fold onto the same face, so one face of the cube is missing and the net fails.It is not a net of a cube. ✓
Check. The same test (fold each square once) shows it fails. ✓
Method. The three rectangles are 3, 4 and 5 cm wide and each is 10 cm long. Put them side by side.
- Add the widths. The widths are the three sides of the triangle.3 + 4 + 5 = 12 cm
- State the size. The rectangles share the same 10 cm length, so together they form one big rectangle.One rectangle 12 cm by 10 cm
Check. 12 × 10 = 120 = 30 + 40 + 50, the three separate areas. ✓
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. It is a right-angled triangle, so use the two shorter sides as base and height: ½ × base × height.½ × 3 × 4 = 6 cm2
- Double it for the two ends. The two end faces are identical.2 × 6 = 12 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the side of length 3: 3 × 10 = 30
Rectangle on the side of length 4: 4 × 10 = 40
Rectangle on the side of length 5: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.12 + 30 + 40 + 50 = 132
- Write the unit. Area, so square units.Surface area = 132 cm2
Check. The rectangles together make (perimeter of an end) × length = 12 × 10 = 120. Then 12 + 120 = 132. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Method. Use the area to find one face, then the edge length, then the volume.
- Find the area of one face. A cube has 6 identical faces.150 ÷ 6 = 25 cm2
- Find the edge length. Each face is a square, so edge × edge = 25. Since 5 × 5 = 25, the edge is 5 cm.Edge = 5 cm
- Find the volume. Volume of a cube = edge × edge × edge.5 × 5 × 5 = 125 cm3
Check. Surface area of a cube with edge 5 is 6 × 25 = 150. ✓
Watch out. Divide by 6 (faces), not by 4 or by 2.
Method. A net has one shape for each face of the solid.
- Count the faces of a cube. A dice has a top, a bottom and four sides.1 + 1 + 4 = 6
- Write the answer. Each face is a square.The net has 6 squares.
Check. The picture of any cube net shows 6 squares. ✓
Method. A cuboid has 6 rectangular faces in 3 matching pairs. Draw them in a cross shape.
- Work out the six rectangles. Two are 3 by 2 (top and bottom), two are 3 by 1 (front and back), two are 2 by 1 (the ends).3 × 2, 3 × 2, 3 × 1, 3 × 1, 2 × 1, 2 × 1
- Draw them joined. Draw a row of four rectangles: 2 by 1, 3 by 1, 2 by 1, 3 by 1. Then add a 3 by 2 rectangle above and one below the 3 by 1 rectangle.See the picture in the answer.
Check. Fold it in your head: the 3 by 2 rectangles become the top and bottom. ✓
Watch out. Edges that join must be the same length.
Method. Think about what you get if you take apart a tin.
- Remove the top and the bottom. These are two circles.2 circles
- Open out the side. The curved side flattens into a rectangle.1 rectangle
Check. 2 circles + 1 rectangle = 3 pieces, as in the picture of the cylinder net. ✓
Plans and elevations
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Imagine walking round a model made of building blocks. From the front it looks one way, from the side another way, and from above another. Architects and engineers draw exactly these views to describe a solid on flat paper.
The view from above is the plan. The view from the front is the front elevation. The view from the side (we use the right-hand side) is the side elevation. Each view is drawn straight on, so you see flat shapes, never a 3D picture. For a solid made of cubes we draw each view on squared paper, one square for each cube face you can see.
How to find an elevation of a stack of cubes. For the front elevation, look at each column from the left to the right and find the tallest stack: a tall stack hides any shorter stack behind it. For the side elevation, look at each row from the front to the back and find its tallest stack. For the plan, just mark every stack as one square (you can write the number of cubes in it).
Other solids. The views of cylinders, cones, spheres, pyramids and prisms are shapes you already know. A cylinder standing on its circular face looks like a rectangle from the front and the side, and like a circle from above. A cone looks like a triangle from the front and a circle (with a dot at the centre) from above.
Key ideas
- Plan: the view from above. Front elevation: the view from the front. Side elevation: the view from the side.
- Draw only what you can see, straight on, with the right sizes on squared paper.
- For cubes: use the tallest stack in each column (front) or row (side). In the side elevation the front is on the left.
- Cylinder: rectangle, rectangle, circle. Cone: triangle, triangle, circle. Sphere: circle, circle, circle. Square-based pyramid: triangle, triangle, square with diagonals.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 1, 3 → tallest = 3
Column 2 (from the left): stacks 2, 1 → tallest = 2 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 3, 1 → tallest = 3
Back row (view position 2): stacks 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Think about what you would see from above and from the front, then draw it to the correct size.
- Plan: look from above. From above you see the circular base, with the point of the cone in the middle.A circle of radius 3 cm with a dot at its centre
- Front elevation: look from the front. From the side-on view the cone looks like a triangle. Its base is the width of the circle and its height is the height of the cone.Base = 2 × 3 = 6 cm, height = 5 cm: an isosceles triangle
- Draw on the grid. Use one square for 1 cm so the drawings are the right size.
Check. The widest part of both drawings is 6 cm, as it should be. ✓
Watch out. From above a cone is a circle, not a triangle.
Method. Draw each view of the triangular prism straight on, using one grid square for 1 cm.
- Front elevation. Looking at the triangular end, we see the triangle: base 4 cm, height 3 cm.Triangle with base 4 and height 3
- Side elevation. From the side we see a rectangular face: as long as the prism (6 cm) and as tall as the triangle (3 cm).Rectangle 6 cm by 3 cm
- Plan. From above we see the two sloping rectangles side by side, each 2 cm wide (half of the base) and 6 cm long. The line between them is the top edge.Rectangle 4 cm by 6 cm with a line down the middle
- Draw on the grid. One square for 1 cm.
Check. The width is 4 in the front elevation and the plan, the height is 3 in both elevations, and the length is 6 in the side elevation and the plan. ✓
Watch out
For the side elevation, the front of the solid is on the left when you look from the right. Do not add the numbers in a column: use the tallest stack, because the shorter stacks are hidden behind it. A plan shows one square for each stack, whatever its height.Exercise 19C Plans and elevations
Non-calculatorKey words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 1, 3 → tallest = 3
Column 2 (from the left): stacks 2, 1 → tallest = 2 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 3, 1 → tallest = 3
Back row (view position 2): stacks 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand above the solid and look straight down. The plan is the view from above. The back row is at the top of the drawing and the front row is at the bottom.
- Mark every position that has at least one cube. From above you see the top of every stack, however tall it is.Row 1 (from the back): 1, 2
Row 2 (from the back): 3, 1 - Draw the outline of each stack. Each stack is one square. Write the number of cubes in it if you want to show the heights.Total number of cubes = 7
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. Count the cubes in the plan numbers: total 7. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 0, 1 → tallest = 1
Column 2 (from the left): stacks 1, 1 → tallest = 1
Column 3 (from the left): stacks 2, 3 → tallest = 3 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 1, 1, 3
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 1 + 1 + 3 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 1, 1, 3 → tallest = 3
Back row (view position 2): stacks 0, 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Add the numbers in all the stacks.
- Read the number in each stack. The back row is 2, 1, 0 and the front row is 1, 3, 1.2, 1, 0 and 1, 3, 1
- Add them. Add the back row, then the front row.(2 + 1 + 0) + (1 + 3 + 1) = 3 + 5 = 8
Check. Add in a different order: 2 + 1 + 1 + 3 + 1 = 8. ✓
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 2, 1 → tallest = 2
Column 2 (from the left): stacks 1, 3 → tallest = 3
Column 3 (from the left): stacks 0, 1 → tallest = 1 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 2, 3, 1
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 6 squares: 2 + 3 + 1 = 6. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 1, 3, 1 → tallest = 3
Back row (view position 2): stacks 2, 1, 0 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A tin stands on its circular base.
- Say what you see. From above you look straight down at the circular lid.The view is a circle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A tin stands on its circular base.
- Say what you see. From the front you see the flat side of the tin: as wide as the circle and as tall as the tin.The view is a rectangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A cone has a point at the top and a circle at the bottom.
- Say what you see. From the front the sloping sides meet at the point.The view is a triangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A ball looks the same from every direction.
- Say what you see. From above, the outline of a ball is a circle.The view is a circle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A pyramid comes to a point at the top.
- Say what you see. From the front you see a triangular face (the base is hidden below).The view is a triangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A cuboid has rectangular faces.
- Say what you see. From above you see the rectangular top face.The view is a rectangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a circle, with a dot where the point is.
- Find the matching picture. The plan of the cone is picture C.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a circle (the lid).
- Find the matching picture. The plan of the cylinder is picture B.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see the square base with lines from each corner to the apex.
- Find the matching picture. The plan of the square-based pyramid is picture A.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a rectangle with the top edge as a line down the middle.
- Find the matching picture. The plan of the triangular prism is picture D.
Check. The other three plans do not match what you would see from above. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The front elevation of a cuboid is a rectangle: as wide as its length and as tall as its height.
- Find the width and height. The 4 cm length faces us and the cuboid is 2 cm high.4 cm wide, 2 cm high
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 8 cm2, the same as a face of 4 by 2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The side elevation is a rectangle: as wide as the cuboid is deep and as tall as its height.
- Find the width and height. From the side we see the 3 cm width, and the height is 2 cm.3 cm wide, 2 cm high
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 6 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The plan is the rectangle you see looking down on the top face.
- Find the sides. The top face is 4 cm long and 3 cm wide.4 cm by 3 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 12 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. From the side we see a rectangle.
- Find its sides. It is as long as the prism and as tall as the triangle.6 cm by 3 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The area is 18 cm2. ✓
Method. The front elevation is the triangle at the end. Use area = ½ × base × height.
- Substitute. Base 4 cm and height 3 cm.Area = ½ × 4 × 3
- Work it out. 4 × 3 = 12, and half of 12 is 6.Area = 6 cm2
Check. A 4 by 3 rectangle has area 12; the triangle is half of it. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A rectangle means the sides are straight up and down.
- Plan. A circle means it is round from above. A rectangle that is round from above is a cylinder.The solid is a cylinder.
Check. A cuboid would have a rectangle for its plan too. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A triangle means it comes to a point at the top.
- Plan. A circle with a dot means a round base and a point above the centre. That is a cone.The solid is a cone.
Check. A pyramid has a square plan, not a circle. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A triangle means it comes to a point at the top.
- Plan. A square with lines from the corners to the middle is the base with the sloping edges going up to the apex.The solid is a square-based pyramid.
Check. A cone has a circle for its plan. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The front elevation of a prism is its cross-section when the end faces you.
- Draw the L-shape. The L is 4 cm wide and 3 cm tall. The bottom arm is 1 cm tall and the upright arm is 1 cm wide.An L made from 4 + 2 = 6 unit squares
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. Area = 4 × 1 + 1 × 2 = 6 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The plan is the view from above: a rectangle with a line where the height changes.
- Find the size. From above the prism is 4 cm wide and 5 cm long.4 cm by 5 cm
- Mark the step. The tall arm is 1 cm wide, so there is a line 1 cm from the left edge.Line at 1 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The plan is a rectangle with area 20 cm2. ✓
Method. Each cube has volume 1 cm3, so the volume is the number of cubes.
- Add the numbers in the plan. Back row 3, 1; middle row 2, 2; front row 1, 1.3 + 1 + 2 + 2 + 1 + 1 = 10 cubes
- Write the volume. 1 cm cube = 1 cm3.Volume = 10 cm3
Check. (3 + 1) + (2 + 2) + (1 + 1) = 4 + 4 + 2 = 10. ✓
Method. The front elevation uses the tallest stack in each column.
- Find the tallest in each column. Left column: 3, 2, 1 → 3. Right column: 1, 2, 1 → 2.Heights 3 and 2
- Add. Each unit of height is one square.3 + 2 = 5
Check. Count the squares in the finished drawing: a column of 3 and a column of 2. ✓
Method. Find the volume of one cube, then multiply by the number of cubes.
- Volume of one cube. Volume = side × side × side.2 × 2 × 2 = 8 cm3
- Multiply by 5. There are five identical cubes.5 × 8 = 40 cm3
Check. The 5 cubes have volume 5 × 8 = 40; no cubes overlap. ✓
Watch out. The cubes have side 2 cm, not 1 cm: each cube is 8 cm3.
Method. Work out the side elevation yourself, then compare it with Mia’s drawing.
- Find the tallest stack in each row. The front row has stacks 3 and 1, so its tallest is 3. The back row has stacks 1 and 2, so its tallest is 2.Front row: 3. Back row: 2.
- Decide where each row goes. Looking from the right, the front is on the left of the view. So the correct side elevation has a column of 3 on the left and a column of 2 on the right.Correct: heights 3 then 2
- Compare. Mia’s drawing has heights 2 then 3: the rows are the wrong way round.Mia is not correct.
Check. She has reversed the rows. Her drawing would be correct for a view from the left. ✓
Watch out. Looking from the right, the front of the solid is on the left.
Method. The plan is one row, so there is only one stack in each column. Each stack is as tall as its column.
- Read the plan. A row of 3 squares means 3 stacks standing side by side, with nothing behind them.3 stacks
- Use the front elevation. Because there is only one stack per column, the heights are 2, 3 and 1.2 + 3 + 1 = 6 cubes
Check. 6 cubes in total; 2 + 3 + 1 = 6. ✓
Method. From the side you see one row only, and the tallest stack is the one that shows.
- Find the tallest stack. The stacks have heights 2, 3 and 1.Tallest = 3
- Describe the view. There is only one row, so the view is one column, 3 squares tall.A column of 3 squares (1 wide, 3 high)
Check. From the side the stack of 3 hides the stacks of 2 and 1. ✓
Method. The plan shows the top of each stack as a square.
- Find the number of squares along each side. The block is 2 wide and 3 long.2 × 3
- Multiply. This counts the squares in a rectangle.2 × 3 = 6
Check. Count by rows: 3 + 3 = 6. ✓
Method. The three views have names that tell you where you are looking from.
- Recall the names. Plan = from above. Front elevation = from the front. Side elevation = from the side.
- Choose. Looking straight down is looking from above.The view is the plan.
Check. Think of a plan of a house: it is drawn from above. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The plan is what you see from above: the circular lid.
- Find the size of the circle. The radius is 4 cm, so the circle is 8 cm across.Radius 4 squares, width 8 squares
- Draw it. Put the centre in the middle of an 8 by 8 square and draw a circle of radius 4 squares.
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The circle touches the edge of the 8 by 8 square at four points. ✓
Volume of cuboids, and units of volume and capacity
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Volume is the amount of space inside a solid. We measure it by counting how many unit cubes fit inside. A cubic centimetre (cm3) is the volume of a cube whose edges are all 1 cm long. A cubic metre (m3) is a cube with 1 m edges, and a cubic millimetre (mm3) a tiny cube with 1 mm edges.
Look at the cuboid below, which is 4 cm long, 3 cm wide and 2 cm high. The bottom layer holds 4 × 3 = 12 cubes. There are 2 layers, so the whole cuboid holds 12 × 2 = 24 cubes: its volume is 24 cm3.
This always works, so the volume of a cuboid is length × width × height. A cube has all edges equal, so its volume is side × side × side. The three lengths must be in the same unit before you multiply.
Capacity. Liquids are measured in litres (l) and millilitres (ml). The capacity of a container is how much it can hold. The links to volume are worth learning:
| Volume | Capacity |
|---|---|
| 1 cm3 | 1 ml |
| 1000 cm3 | 1 litre (1000 ml) |
| 1 m3 = 1,000,000 cm3 | 1000 litres |
| 1 cm3 | 1000 mm3 |
Why is 1 m3 equal to 1,000,000 cm3? Because 1 m = 100 cm, so a cube of side 1 m has volume 100 × 100 × 100 = 1,000,000 cm3. In the same way 1 cm = 10 mm, so 1 cm3 = 10 × 10 × 10 = 1000 mm3. Volume units are not like length units: you cube the conversion factor.
Key ideas
- Volume of a cuboid = length × width × height. Volume of a cube = side3.
- Volume is measured in cubic units: mm3, cm3, m3.
- 1 ml = 1 cm3. 1 litre = 1000 ml = 1000 cm3. 1 m3 = 1000 litres = 1,000,000 cm3.
- To find a missing length, divide the volume by the other two lengths.
- Put all lengths in the same unit before you multiply.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 8 × 5 × 4
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.8 × 5 = 40
- Multiply by the height. This counts all the layers.40 × 4 = 160
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 160 cm3
Check. Multiply in a different order: 5 × 4 = 20, then 20 × 8 = 160. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Method. Use Volume = length × width × height with the unknown length as a letter, then divide to find it.
- Write the volume formula with the numbers you know. Volume = length × width × height. We call the missing length h.240 = 8 × 5 × h
- Multiply the two known lengths. Join the known numbers into one number.8 × 5 = 40, so 240 = 40 × h
- Divide to undo the multiplication. Dividing is the inverse of multiplying.h = 240 ÷ 40 = 6
- Write the unit. This is a length, so it is in cm (not cm3).h = 6 cm
Check. 8 × 5 × 6 = 240. ✓
Watch out. Divide the volume by the two known lengths, not by just one of them.
Method. Find the volume in cm3, then divide by 1000 to change cm3 into litres.
- Work out the volume of the cuboid. Volume = length × width × height, all in cm.80 × 50 = 4000
4000 × 40 = 160,000 cm3 - Change cm3 to litres. 1 litre = 1000 cm3, so divide by 1000.160,000 ÷ 1000 = 160
- Write the answer. Capacity = 160 litres
Check. 160 litres × 1000 = 160,000 cm3. ✓
Watch out. 1 litre is 1000 cm3. Do not divide by 100.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change m3 into cm3. 1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3.2.5 m3 = 2.5 × 1,000,000 = 2,500,000 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.2,500,000 cm3 = 2,500,000 ÷ 1000 = 2500 litres
- Write the answer. Check that the unit matches what the question asked for.2.5 m3 = 2500 litres
Check. Go back the other way: 2500 litres gives 2.5 m3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Method. A cube of side a has volume a × a × a. To find the side, find the number that multiplies by itself three times to give the volume (the cube root).
- Write the volume as a cube. We are looking for a number a with a × a × a = the volume.a × a × a = 343
- Try whole numbers. Test cubes you know: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.7 × 7 × 7 = 49 × 7 = 343
- State the side length. The side is the cube root of the volume.a = 7 cm
Check. 7 × 7 × 7 = 343. ✓
Watch out. The cube root is not the volume divided by 3.
Watch out
Volume is in cubic units (cm3), so do not write cm or cm2. Do not add the lengths: multiply them. And do not change m3 to cm3 by multiplying by 100: you must multiply by 100 × 100 × 100 = 1,000,000.Exercise 19D Volume of cuboids and units of volume
Non-calculatorMethod. Count the cubes in one layer: they form a rectangle 4 by 3.
- Find the sides of the layer. The layer is 4 cubes long and 3 cubes wide.4 by 3
- Multiply. Rows × cubes in each row.4 × 3 = 12
Check. Count the cubes in 3 rows of 4: 4 + 4 + 4 = 12. ✓
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 4 × 3 × 2
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.4 × 3 = 12
- Multiply by the height. This counts all the layers.12 × 2 = 24
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 24 cm3
Check. Multiply in a different order: 3 × 2 = 6, then 6 × 4 = 24. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 5 × 4 × 3
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.5 × 4 = 20
- Multiply by the height. This counts all the layers.20 × 3 = 60
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 60 cm3
Check. Multiply in a different order: 4 × 3 = 12, then 12 × 5 = 60. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 10 × 6 × 2
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.10 × 6 = 60
- Multiply by the height. This counts all the layers.60 × 2 = 120
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 120 cm3
Check. Multiply in a different order: 6 × 2 = 12, then 12 × 10 = 120. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 2.5 × 4 × 6
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.2.5 × 4 = 10
- Multiply by the height. This counts all the layers.10 × 6 = 60
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 60 cm3
Check. Multiply in a different order: 4 × 6 = 24, then 24 × 2.5 = 60. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in m, so the units match.V = 3 × 2 × 1.5
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.3 × 2 = 6
- Multiply by the height. This counts all the layers.6 × 1.5 = 9
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 9 m3
Check. Multiply in a different order: 2 × 1.5 = 3, then 3 × 3 = 9. ✓
Watch out. Volume is in m3, not m or m2. Do not add the lengths.
Method. A cube has three equal lengths, so Volume = side × side × side.
- Write down the formula. Length, width and height of a cube are all the same size.V = side × side × side
- Substitute the side length. V = 6 × 6 × 6
- Multiply two of them first. Do it in two small steps.6 × 6 = 36
- Multiply by the last one. 36 × 6 = 216
- Write the unit. Cubic units.V = 216 cm3
Check. 63 on a calculator gives 216. ✓
Watch out. 63 means 6 × 6 × 6, not 6 × 3.
Method. A cube has three equal lengths, so Volume = side × side × side.
- Write down the formula. Length, width and height of a cube are all the same size.V = side × side × side
- Substitute the side length. V = 9 × 9 × 9
- Multiply two of them first. Do it in two small steps.9 × 9 = 81
- Multiply by the last one. 81 × 9 = 729
- Write the unit. Cubic units.V = 729 cm3
Check. 93 on a calculator gives 729. ✓
Watch out. 93 means 9 × 9 × 9, not 9 × 3.
Method. A cube has three equal lengths, so Volume = side × side × side.
- Write down the formula. Length, width and height of a cube are all the same size.V = side × side × side
- Substitute the side length. V = 20 × 20 × 20
- Multiply two of them first. Do it in two small steps.20 × 20 = 400
- Multiply by the last one. 400 × 20 = 8000
- Write the unit. Cubic units.V = 8000 mm3
Check. 203 on a calculator gives 8000. ✓
Watch out. 203 means 20 × 20 × 20, not 20 × 3.
Method. Use Volume = length × width × height with the unknown length as a letter, then divide to find it.
- Write the volume formula with the numbers you know. Volume = length × width × height. We call the missing length h.240 = 8 × 5 × h
- Multiply the two known lengths. Join the known numbers into one number.8 × 5 = 40, so 240 = 40 × h
- Divide to undo the multiplication. Dividing is the inverse of multiplying.h = 240 ÷ 40 = 6
- Write the unit. This is a length, so it is in cm (not cm3).h = 6 cm
Check. 8 × 5 × 6 = 240. ✓
Watch out. Divide the volume by the two known lengths, not by just one of them.
Method. Use Volume = length × width × height with the unknown length as a letter, then divide to find it.
- Write the volume formula with the numbers you know. Volume = length × width × height. We call the missing length w.180 = 6 × 5 × w
- Multiply the two known lengths. Join the known numbers into one number.6 × 5 = 30, so 180 = 30 × w
- Divide to undo the multiplication. Dividing is the inverse of multiplying.w = 180 ÷ 30 = 6
- Write the unit. This is a length, so it is in cm (not cm3).w = 6 cm
Check. 6 × 5 × 6 = 180. ✓
Watch out. Divide the volume by the two known lengths, not by just one of them.
Method. Use Volume = length × width × height with the unknown length as a letter, then divide to find it.
- Write the volume formula with the numbers you know. Volume = length × width × height. We call the missing length l.96 = 4 × 3 × l
- Multiply the two known lengths. Join the known numbers into one number.4 × 3 = 12, so 96 = 12 × l
- Divide to undo the multiplication. Dividing is the inverse of multiplying.l = 96 ÷ 12 = 8
- Write the unit. This is a length, so it is in cm (not cm3).l = 8 cm
Check. 4 × 3 × 8 = 96. ✓
Watch out. Divide the volume by the two known lengths, not by just one of them.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change litres into cm3. 1 litre = 1000 ml = 1000 cm3.3 litres = 3 × 1000 = 3000 cm3
- Change cm3 into ml. 1 millilitre (ml) is the same amount as 1 cm3.3000 cm3 = 3000 ml
- Write the answer. Check that the unit matches what the question asked for.3 litres = 3000 ml
Check. Go back the other way: 3000 ml gives 3 litres. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change ml into cm3. 1 millilitre (ml) is the same amount as 1 cm3.4500 ml = 4500 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.4500 cm3 = 4500 ÷ 1000 = 4.5 litres
- Write the answer. Check that the unit matches what the question asked for.4500 ml = 4.5 litres
Check. Go back the other way: 4.5 litres gives 4500 ml. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Decide which conversion fact we need. Every volume unit can be linked to cm3.1 litre = 1000 ml = 1000 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.2000 cm3 = 2000 ÷ 1000 = 2 litres
- Write the answer. Check that the unit matches what the question asked for.2000 cm3 = 2 litres
Check. Go back the other way: 2 litres gives 2000 cm3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change m3 into cm3. 1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3.0.5 m3 = 0.5 × 1,000,000 = 500,000 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.500,000 cm3 = 500,000 ÷ 1000 = 500 litres
- Write the answer. Check that the unit matches what the question asked for.0.5 m3 = 500 litres
Check. Go back the other way: 500 litres gives 0.5 m3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Decide which conversion fact we need. Every volume unit can be linked to cm3.1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3
- Change m3 into cm3. 1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3.5 m3 = 5 × 1,000,000 = 5,000,000 cm3
- Write the answer. Check that the unit matches what the question asked for.5 m3 = 5,000,000 cm3
Check. Go back the other way: 5,000,000 cm3 gives 5 m3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Decide which conversion fact we need. Every volume unit can be linked to cm3.1 cm = 10 mm, so 1 cm3 = 10 × 10 × 10 = 1000 mm3
- Change mm3 into cm3. 1 cm = 10 mm, so 1 cm3 = 10 × 10 × 10 = 1000 mm3.7000 mm3 = 7000 ÷ 1000 = 7 cm3
- Write the answer. Check that the unit matches what the question asked for.7000 mm3 = 7 cm3
Check. Go back the other way: 7 cm3 gives 7000 mm3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Method. Find the volume in cm3, then divide by 1000 to change cm3 into litres.
- Work out the volume of the cuboid. Volume = length × width × height, all in cm.60 × 30 = 1800
1800 × 40 = 72,000 cm3 - Change cm3 to litres. 1 litre = 1000 cm3, so divide by 1000.72,000 ÷ 1000 = 72
- Write the answer. Capacity = 72 litres
Check. 72 litres × 1000 = 72,000 cm3. ✓
Watch out. 1 litre is 1000 cm3. Do not divide by 100.
Method. Find the capacity of the tank in litres, then divide by the rate of filling.
- Find the volume in cm3. Volume = length × width × height.80 × 50 = 4000, then 4000 × 40 = 160,000 cm3
- Change to litres. 1 litre = 1000 cm3, so divide by 1000.160,000 ÷ 1000 = 160 litres
- Divide by the rate. Each minute 8 litres go in, so count how many 8s make 160.160 ÷ 8 = 20 minutes
Check. In 20 minutes, 20 × 8 = 160 litres go in. ✓
Watch out. Change cm3 to litres before you use the rate in litres.
Method. Find how many cubes fit along each edge, then multiply. (Divide the volumes only if the cubes fit exactly.)
- Cubes along the length. Divide the length by the side of the cube.20 ÷ 5 = 4
- Cubes along the width. 15 ÷ 5 = 3
- Cubes up the height. 10 ÷ 5 = 2
- Multiply. This counts the cubes in one layer, then all the layers.4 × 3 × 2 = 24
Check. Volume of the box 20 × 15 × 10 = 3000; one cube is 5 × 5 × 5 = 125; 3000 ÷ 125 = 24. ✓
Watch out. Dividing the volumes works only when the cubes fit with no gaps.
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in m, so the units match.V = 10 × 4 × 1.5
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.10 × 4 = 40
- Multiply by the height. This counts all the layers.40 × 1.5 = 60
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 60 m3
Check. Multiply in a different order: 4 × 1.5 = 6, then 6 × 10 = 60. ✓
Watch out. Volume is in m3, not m or m2. Do not add the lengths.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change m3 into cm3. 1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3.60 m3 = 60 × 1,000,000 = 60,000,000 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.60,000,000 cm3 = 60,000,000 ÷ 1000 = 60,000 litres
- Write the answer. Check that the unit matches what the question asked for.60 m3 = 60,000 litres
Check. Go back the other way: 60,000 litres gives 60 m3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Method. Work out both volumes, compare them, then find the difference.
- Volume of box A. length × width × height.30 × 20 = 600, then 600 × 10 = 6000 cm3
- Volume of box B. 25 × 25 = 625, then 625 × 10 = 6250 cm3
- Compare and subtract. Box B is bigger. Subtract the smaller from the larger.6250 − 6000 = 250
- Write the answer. Box B is larger, by 250 cm3.
Check. B is 625 × 10, A is 600 × 10, so B is 25 × 10 = 250 bigger. ✓
Method. Check which operation the volume formula uses, then redo the calculation.
- Spot the mistake. Volume = length × width × height. Tom added the lengths, which would give something like the total of three edges, not a volume.Tom added instead of multiplying.
- Do it correctly. Multiply the first two, then the third.6 × 4 = 24, then 24 × 3 = 72
- Write the answer with its unit. Volume = 72 cm3
Check. 3 layers of 24 cubes: 3 × 24 = 72. ✓
Watch out. Volume needs multiplication, not addition.
Method. Find whole numbers that multiply to give 36. Start with a factor pair of 36 and split one factor again.
- List the factors of 36. The edges must be factors of 36.1, 2, 3, 4, 6, 9, 12, 18, 36
- Choose three that multiply to 36. Try combinations and check the product.36 × 1 × 1 = 36
18 × 2 × 1 = 36
12 × 3 × 1 = 36
9 × 4 × 1 = 36
9 × 2 × 2 = 36
6 × 6 × 1 = 36
6 × 3 × 2 = 36
4 × 3 × 3 = 36 - Write three of them. Any three of these sets are correct.36 × 1 × 1, 18 × 2 × 1, 12 × 3 × 1
Check. Each product is 36, and each set uses whole numbers. ✓
Method. A cubic metre is a cube. Work out how many 1 cm cubes it holds.
- Think of a cube of side 1 m. 1 m = 100 cm, so the cube is 100 cm long, 100 cm wide and 100 cm high.100 cm × 100 cm × 100 cm
- Work out its volume in cm3. Volume = length × width × height.100 × 100 = 10,000, then 10,000 × 100 = 1,000,000
- Compare with Jo. Jo multiplied the conversion factor once instead of three times.1 m3 = 1,000,000 cm3, not 100 cm3.
Check. 1 litre = 1000 cm3 and 1 m3 = 1000 litres, so 1000 × 1000 = 1,000,000. ✓
Method. A cube has three equal lengths, so Volume = side × side × side.
- Write down the formula. Length, width and height of a cube are all the same size.V = side × side × side
- Substitute the side length. V = 2 × 2 × 2
- Multiply two of them first. Do it in two small steps.2 × 2 = 4
- Multiply by the last one. 4 × 2 = 8
- Write the unit. Cubic units.V = 8 cm3
Check. 23 on a calculator gives 8. ✓
Watch out. 23 means 2 × 2 × 2, not 2 × 3.
Method. Think about how many lengths are multiplied to get each measurement.
- Length. A length is one measurement along a line.cm
- Area. Area multiplies two lengths, for example length × width.cm × cm = cm2
- Volume. Volume multiplies three lengths.cm × cm × cm = cm3
Check. The little number tells you how many lengths were multiplied. ✓
Method. Recall the two facts that link capacity and volume.
- Litres and millilitres. Milli means one thousandth.1 litre = 1000 ml
- Millilitres and cm3. 1 ml and 1 cm3 are the same amount.1000 ml = 1000 cm3
Check. A 1 litre carton is 10 cm by 10 cm by 10 cm: 10 × 10 × 10 = 1000 cm3. ✓
Volume of prisms
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A prism is a solid that has the same cross-section all the way along its length. Cut a loaf of bread into slices: every slice is identical. A cuboid, a triangular tent, a Toblerone box and a trough are all prisms. The two identical faces at the ends are the cross-section, and the distance between them is the length of the prism.
Here is why there is one simple rule. Imagine the cross-section as a thin slice. The volume of the slice is its area × its thickness. Stack up slices all along the length, and you get the whole prism. So:
Volume of a prism = area of cross-section × length
The skill is finding the area of the cross-section. Remember the area formulas from Chapter 18:
| Shape | Area |
|---|---|
| rectangle | width × height |
| triangle | ½ × base × perpendicular height |
| parallelogram | base × perpendicular height |
| trapezium | ½ × (sum of the parallel sides) × the distance between them |
| L-shape or other compound shape | split into rectangles and add |
Which length is the “length”? A prism can lie on any face. Always find the cross-section first: it is the face that is repeated. The “length” is the distance between the two copies of it. If a triangular prism stands on one of its triangles, the “length” is its height!
Key ideas
- Volume of a prism = area of cross-section × length.
- Find the area of the cross-section first, using the perpendicular height.
- The length is the distance between the two identical end faces.
- To find the length, divide the volume by the area of the cross-section.
- Volume is in cubic units: cm3, m3.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: triangle. Length = 10 cm
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 6 × 4 = 24 ÷ 2 = 12 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 12 × 10 = 120
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 120 cm3
Check. A triangular prism is half of a cuboid 6 × 4 × 10 = 240, and half of that is 120. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: trapezium. Length = 12 cm
- Work out the area of the trapezium cross-section. Area of a trapezium = ½ × (sum of the parallel sides) × the distance between them.A = ½ × (4 + 8) × 3
A = ½ × 12 × 3 = 36 ÷ 2 = 18 cm2 - Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 18 × 12 = 216
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 216 cm3
Check. Estimate: the volume must be between 144 and 288 (the volumes of the prisms using only the shorter or only the longer parallel side), and 216 is in that range. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: L-shape. Length = 10 cm
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 28 × 10 = 280
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 280 cm3
Check. Split the L the other way: left rectangle 3 × 6 = 18, right rectangle 5 × 2 = 10; total 28 = 28. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Method. Work backwards: the volume divided by the length gives the area of the cross-section. Then use the triangle area formula to find its height.
- Use V = area × length to find the cross-section area. Divide both sides by the length.180 = A × 15
A = 180 ÷ 15 = 12 cm2 - Write the triangle area formula. Area of a triangle = ½ × base × height.12 = ½ × 6 × h
- Simplify the right-hand side. ½ × 6 = 3.12 = 3 × h
- Divide to find h. Undo the multiplication.h = 12 ÷ 3 = 4 cm
Check. ½ × 6 × 4 = 12, and 12 × 15 = 180. ✓
Watch out. The volume is not the area of the triangle. Divide by the length first to get the area.
Method. Find the volume in cm3 using area of cross-section × length, then change to litres.
- Area of the triangular cross-section. Area = ½ × base × perpendicular height.½ × 40 × 30 = 600 cm2
- Multiply by the length. Volume = cross-section × length.600 × 100 = 60,000 cm3
- Change to litres. 1 litre = 1000 cm3, so divide by 1000.60,000 ÷ 1000 = 60 litres
Check. 60 litres × 1000 = 60,000 cm3. ✓
Watch out. Change units only at the end, once the volume is found in cm3.
Watch out
Use the perpendicular height of the triangle or trapezium, not the sloping side. And remember to multiply by the length at the end: finding only the area of the cross-section is not enough. Check the units: area is in cm2, volume in cm3.Exercise 19E Volume of prisms
Non-calculatorKey words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: triangle. Length = 10 cm
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 6 × 4 = 24 ÷ 2 = 12 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 12 × 10 = 120
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 120 cm3
Check. A triangular prism is half of a cuboid 6 × 4 × 10 = 240, and half of that is 120. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: triangle. Length = 8 cm
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 5 × 12 = 60 ÷ 2 = 30 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 30 × 8 = 240
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 240 cm3
Check. A triangular prism is half of a cuboid 5 × 12 × 8 = 480, and half of that is 240. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: triangle. Length = 5 cm
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 9 × 4 = 36 ÷ 2 = 18 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 18 × 5 = 90
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 90 cm3
Check. A triangular prism is half of a cuboid 9 × 4 × 5 = 180, and half of that is 90. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: trapezium. Length = 12 cm
- Work out the area of the trapezium cross-section. Area of a trapezium = ½ × (sum of the parallel sides) × the distance between them.A = ½ × (4 + 8) × 3
A = ½ × 12 × 3 = 36 ÷ 2 = 18 cm2 - Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 18 × 12 = 216
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 216 cm3
Check. Estimate: the volume must be between 144 and 288 (the volumes of the prisms using only the shorter or only the longer parallel side), and 216 is in that range. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: trapezium. Length = 10 cm
- Work out the area of the trapezium cross-section. Area of a trapezium = ½ × (sum of the parallel sides) × the distance between them.A = ½ × (5 + 9) × 4
A = ½ × 14 × 4 = 56 ÷ 2 = 28 cm2 - Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 28 × 10 = 280
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 280 cm3
Check. Estimate: the volume must be between 200 and 360 (the volumes of the prisms using only the shorter or only the longer parallel side), and 280 is in that range. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: parallelogram. Length = 9 cm
- Work out the area of the parallelogram cross-section. Area of a parallelogram = base × perpendicular height (not the slanted side).A = 6 × 4 = 24 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 24 × 9 = 216
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 216 cm3
Check. It has the same volume as a cuboid 6 × 4 × 9 = 216. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: L-shape. Length = 10 cm
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 28 × 10 = 280
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 280 cm3
Check. Split the L the other way: left rectangle 3 × 6 = 18, right rectangle 5 × 2 = 10; total 28 = 28. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: L-shape. Length = 4 cm
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 10 by 3
Top rectangle: 2 by (5 − 3) = 2 by 2 - Find the area of each rectangle. Width × height each time.10 × 3 = 30 cm2
2 × 2 = 4 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 30 + 4 = 34 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 34 × 4 = 136
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 136 cm3
Check. Split the L the other way: left rectangle 2 × 5 = 10, right rectangle 8 × 3 = 24; total 34 = 34. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Method. Find the area of the cross-section first, then divide the volume by it to get the length of the prism.
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 6 × 4 = 24 ÷ 2 = 12 cm2
- Write the prism volume formula with the unknown length. Volume = area of cross-section × length.180 = 12 × L
- Divide by the area. Dividing undoes the multiplication.L = 180 ÷ 12 = 15
- Write the unit. A length.Length = 15 cm
Check. 12 × 15 = 180. ✓
Watch out. Do the area of the cross-section first. Do not divide the volume by one of the end lengths.
Method. Find the area of the cross-section first, then divide the volume by it to get the length of the prism.
- Work out the area of the trapezium cross-section. Area of a trapezium = ½ × (sum of the parallel sides) × the distance between them.A = ½ × (4 + 8) × 3
A = ½ × 12 × 3 = 36 ÷ 2 = 18 cm2 - Write the prism volume formula with the unknown length. Volume = area of cross-section × length.216 = 18 × L
- Divide by the area. Dividing undoes the multiplication.L = 216 ÷ 18 = 12
- Write the unit. A length.Length = 12 cm
Check. 18 × 12 = 216. ✓
Watch out. Do the area of the cross-section first. Do not divide the volume by one of the end lengths.
Method. Work backwards: the volume divided by the length gives the area of the cross-section. Then use the triangle area formula to find its height.
- Use V = area × length to find the cross-section area. Divide both sides by the length.150 = A × 10
A = 150 ÷ 10 = 15 cm2 - Write the triangle area formula. Area of a triangle = ½ × base × height.15 = ½ × 6 × h
- Simplify the right-hand side. ½ × 6 = 3.15 = 3 × h
- Divide to find h. Undo the multiplication.h = 15 ÷ 3 = 5 cm
Check. ½ × 6 × 5 = 15, and 15 × 10 = 150. ✓
Watch out. The volume is not the area of the triangle. Divide by the length first to get the area.
Method. Work backwards: the volume divided by the length gives the area of the cross-section. Then use the triangle area formula to find its height.
- Use V = area × length to find the cross-section area. Divide both sides by the length.160 = A × 5
A = 160 ÷ 5 = 32 cm2 - Write the triangle area formula. Area of a triangle = ½ × base × height.32 = ½ × 8 × h
- Simplify the right-hand side. ½ × 8 = 4.32 = 4 × h
- Divide to find h. Undo the multiplication.h = 32 ÷ 4 = 8 cm
Check. ½ × 8 × 8 = 32, and 32 × 5 = 160. ✓
Watch out. The volume is not the area of the triangle. Divide by the length first to get the area.
Method. Find the volume of the tent, then divide by the air each person needs. Round down because you cannot have part of a person.
- Area of the triangular end. ½ × base × height.½ × 3 × 2 = 3 m2
- Volume of the tent. Area of cross-section × length.3 × 4 = 12 m3
- Divide by 2.5 m3 each. This counts how many people fit.12 ÷ 2.5 = 4.8
- Round down. 4.8 people means 4 whole people, with air left over.4 people
Check. 4 people need 4 × 2.5 = 10 m3, which is less than 12. 5 people need 12.5, which is more than 12. ✓
Watch out. Round down when the answer is a number of whole objects that must fit.
Method. Find the volume in cm3, then change to litres.
- Area of the triangle. ½ × base × perpendicular height.½ × 40 × 30 = 600 cm2
- Volume. Cross-section × length.600 × 100 = 60,000 cm3
- Change to litres. 1 litre = 1000 cm3.60,000 ÷ 1000 = 60 litres
Check. 60 × 1000 = 60,000. ✓
Watch out. Do not mix cm and m in the same calculation.
Method. The side of the pool is the trapezium cross-section: the parallel sides are the two depths (1 m and 3 m) and the distance between them is the 20 m length. The prism length is the 8 m width.
- Area of the trapezium. Area = ½ × (sum of the parallel sides) × distance between them.½ × (1 + 3) × 20 = ½ × 4 × 20 = 40 m2
- Multiply by the width of the pool. This is the length of the prism.40 × 8 = 320 m3
Check. Average depth (1 + 3) ÷ 2 = 2 m, and 20 × 8 × 2 = 320. ✓
Watch out. The parallel sides here are the depths, not the length of the pool.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Change m3 into cm3. 1 m = 100 cm, so 1 m3 = 100 × 100 × 100 = 1,000,000 cm3.320 m3 = 320 × 1,000,000 = 320,000,000 cm3
- Change cm3 into litres. 1 litre = 1000 ml = 1000 cm3. To go to the bigger unit, divide.320,000,000 cm3 = 320,000,000 ÷ 1000 = 320,000 litres
- Write the answer. Check that the unit matches what the question asked for.320 m3 = 320,000 litres
Check. Go back the other way: 320,000 litres gives 320 m3. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Method. Work out both volumes and compare them. In a “show that” question, write both calculations and a conclusion.
- Volume of the triangular prism. Area of cross-section = ½ × 8 × 6 = 24 cm2, then multiply by the length.24 × 10 = 240 cm3
- Volume of the cuboid. length × width × height.5 × 6 = 30, then 30 × 8 = 240 cm3
- Compare. Both volumes are 240 cm3, so they are the same. ✓
Check. Different shapes can have the same volume. ✓
Method. Check the area of the cross-section. It is a triangle, not a rectangle.
- Spot the mistake. Sam used base × height, which is the area of a rectangle. A triangle has half that area.Sam forgot the ½.
- Correct area. ½ × 6 × 4.½ × 24 = 12 cm2
- Correct volume. Area × length.12 × 10 = 120 cm3
Check. Sam’s answer, 240, is the volume of the cuboid 6 by 4 by 10; the prism is half of it. ✓
Watch out. Triangle area always has a ½.
Method. Write the volume formula and change L.
- Write the formula. Area × length.V = ½ × b × h × L
- Replace L by 2L. The 2 can be moved to the front.½bh × 2L = 2 × (½bhL) = 2V
Check. A prism twice as long holds twice as much. ✓
Method. Write the volume formula and change b.
- Write the formula. V = ½ × b × h × L
- Replace b by 2b. The 2 can be moved to the front.½ × 2b × h × L = 2 × (½bhL) = 2V
Check. A triangle with twice the base has twice the area. ✓
Method. Write the volume formula and change b and h.
- Write the formula. V = ½ × b × h × L
- Replace b by 2b and h by 2h. Two 2s multiply to 4.½ × 2b × 2h × L = 4 × (½bhL) = 4V
Check. The triangle’s area is multiplied by 2 × 2 = 4. ✓
Method. Find the area of the cross-section first, then divide the volume by it to get the length of the prism.
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 9 by 2
Top rectangle: 3 by (7 − 2) = 3 by 5 - Find the area of each rectangle. Width × height each time.9 × 2 = 18 cm2
3 × 5 = 15 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 18 + 15 = 33 cm2
- Write the prism volume formula with the unknown length. Volume = area of cross-section × length.330 = 33 × L
- Divide by the area. Dividing undoes the multiplication.L = 330 ÷ 33 = 10
- Write the unit. A length.Length = 10 cm
Check. 33 × 10 = 330. ✓
Watch out. Do the area of the cross-section first. Do not divide the volume by one of the end lengths.
Method. Volume of a prism = area of cross-section × length. The area is given, so no area work is needed.
- Write the formula and substitute. Volume = 15 × 6
- Multiply. 10 × 6 = 60 and 5 × 6 = 30.60 + 30 = 90
- Write the unit. 90 cm3
Check. 6 × 15 = 90. ✓
Method. Area of a triangle = ½ × base × perpendicular height.
- Substitute. Area = ½ × 6 × 4
- Multiply, then halve. 6 × 4 = 24 and half of 24 is 12.Area = 12 cm2
Check. A 6 by 4 rectangle has area 24; the triangle is half. ✓
Method. A prism with a rectangle as its cross-section is a cuboid. Use cross-section × length.
- Find the area of the rectangle. width × height.5 × 4 = 20 cm2
- Multiply by the length. 20 × 3 = 60 cm3
Check. 5 × 4 × 3 = 60 as a cuboid. ✓
Volume of cylinders, and density
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A cylinder is a prism with a circle as its cross-section. So the volume rule for prisms still works:
Volume of a cylinder = area of the circular end × height = πr2h
Here r is the radius of the circle (half the diameter) and h is the height (the distance between the two circular ends). You met the area of a circle, πr2, in Chapter 18. For a cylinder with radius 4 cm and height 10 cm the area of the end is π × 42 = 16π cm2, and so the volume is 16π × 10 = 160π cm3.
Exact answers and decimal answers. When a question says “give your answer in terms of π”, leave the π symbol in the answer: 160π cm3. This is the exact answer. When it says “give your answer to 3 significant figures” (or 1 decimal place), use the π button on a calculator: 160π = 502.65… = 503 cm3 to 3 s.f. Never use 3.14 unless the question tells you to.
Density. Lead is heavy and cork is light, even for the same size piece. Density measures how much mass is in each unit of volume: for example, steel has a density of about 7.8 g/cm3, so every cm3 of steel has a mass of 7.8 g. Density, mass and volume are linked by
density = mass ÷ volume, mass = density × volume, volume = mass ÷ density
Always keep the units together: a mass in grams and a volume in cm3 give a density in g/cm3; a mass in kg and a volume in m3 give a density in kg/m3. Compound measures are covered in more depth in Chapter 16.
Key ideas
- Volume of a cylinder = πr2h, where r is the radius. If you are given the diameter, halve it.
- “In terms of π” means leave π in the answer: for example 160π.
- For a decimal answer use the π button, and only round at the end.
- Density = mass ÷ volume. Mass = density × volume. Volume = mass ÷ density.
- To find h from the volume, divide the volume by πr2.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 42 × 10
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.42 = 4 × 4 = 16
- Multiply by the height. 16 × 10 = 160, so V = 160π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 160π cm3
Check. Estimate with π ≈ 3: 3 × 16 × 10 = 480, which is close to 503. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Find the radius. The formulas use the radius, which is half the diameter.r = 10 ÷ 2 = 5
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 52 × 6
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.52 = 5 × 5 = 25
- Multiply by the height. 25 × 6 = 150, so V = 150π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 52 × 6 = 471.2389…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.471.2389… = 471 cm3
Check. Estimate with π ≈ 3: 3 × 25 × 6 = 450, which is close to 471. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.
- Write the volume formula with the numbers. We know V and r. The height h is unknown.180π = π × 62 × h
- Work out r2. 62 = 36, so 180π = 36π × h
- Divide both sides by π. This cancels the π on each side.180 = 36 × h
- Divide by the number in front of h. Undo the multiplication.h = 180 ÷ 36 = 5 cm
Check. π × 62 × 5 = 180π. ✓
Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.
Method. Find the volume in cm3 using πr2h, then change cm3 to litres by dividing by 1000.
- Find the radius. The formula uses the radius, which is half of the diameter.r = 8 ÷ 2 = 4 cm
- Write the volume formula and substitute. V = π × 42 × 12 = π × 16 × 12 = 192π
- Use your calculator. Use the π button and keep the whole display.192π = 603.1858… cm3
- Change to litres. 1 litre = 1000 cm3, so divide by 1000.603.1858… ÷ 1000 = 0.60319… litres
- Round to 2 decimal places. The third decimal place decides whether the second goes up.0.60 litres
Check. Estimate: 3 × 16 × 12 = 576 cm3, so about 0.6 litres. ✓
Watch out. Divide by 1000 to change cm3 to litres. Do not round before the last step.
Method. A semicircle is half a circle, so the prism is half of a cylinder with the same radius and length.
- Volume of the full cylinder. V = πr2h with r = 4 and h = 10 (the length).π × 42 × 10 = 160π
- Halve it. The prism is half of the cylinder.160π ÷ 2 = 80π
- Write the unit. 80π cm3
Check. Area of a semicircle = ½πr2 = 8π, and 8π × 10 = 80π. ✓
Watch out. Halve the volume of the cylinder; do not halve the radius.
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Density = mass ÷ volume.
- Write down the formula. Density is mass shared out over each unit of volume.Density = mass ÷ volume
- Substitute. Mass is in g and volume in cm3, so the answer will be in g/cm3.Density = 450 ÷ 50
- Divide. 450 ÷ 50 = 9
- Write the unit. Density = 9 g/cm3
Check. Mass = density × volume = 9 × 50 = 450 g. ✓
Watch out. Divide the mass by the volume, not the other way round.
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Mass = density × volume.
- Write down the formula. Rearrange Density = mass ÷ volume by multiplying both sides by the volume.Mass = density × volume
- Substitute. Density is 750 kg/m3: every m3 has a mass of 750 kg.Mass = 750 × 2
- Multiply. 750 × 2 = 1500
- Write the unit. Mass = 1500 kg
Check. 1500 ÷ 2 = 750, the density we started with. ✓
Watch out. Check that the volume is in the same unit as in the density.
Method. First find the volume of the cylinder, then use mass = density × volume.
- Volume of the cylinder. V = πr2h.V = π × 22 × 5 = 20π cm3
- Use the calculator for the volume. Keep the whole display.20π = 62.8319… cm3
- Use mass = density × volume. Every cm3 has a mass of 8 g.8 × 62.8319… = 502.655… g
- Round to the nearest gram. 503 g
Check. Exact: mass = 8 × 20π = 160π g, and 160π is about 502.65. ✓
Watch out. Find the volume first. Do not multiply the density by the radius or the height.
Watch out
Use the radius, not the diameter, in πr2h. If the diameter is given, halve it first. Square only the radius. And for density, make sure the volume is in the same unit as in the density (cm3 with g/cm3, m3 with kg/m3).Exercise 19F Cylinders and density: exact answers (non-calculator)
Non-calculatorKey words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 32 × 10
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.32 = 3 × 3 = 9
- Multiply by the height. 9 × 10 = 90, so V = 90π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 90π cm3
Check. Estimate with π ≈ 3: 3 × 9 × 10 = 270, which is close to 283. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 52 × 4
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.52 = 5 × 5 = 25
- Multiply by the height. 25 × 4 = 100, so V = 100π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 100π cm3
Check. Estimate with π ≈ 3: 3 × 25 × 4 = 300, which is close to 314. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Find the radius. The formulas use the radius, which is half the diameter.r = 8 ÷ 2 = 4
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 42 × 5
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.42 = 4 × 4 = 16
- Multiply by the height. 16 × 5 = 80, so V = 80π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 80π cm3
Check. Estimate with π ≈ 3: 3 × 16 × 5 = 240, which is close to 251. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 1.52 × 8
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.1.52 = 1.5 × 1.5 = 2.25
- Multiply by the height. 2.25 × 8 = 18, so V = 18π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 18π cm3
Check. Estimate with π ≈ 3: 3 × 2.25 × 8 = 54, which is close to 57. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 62 × 7
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.62 = 6 × 6 = 36
- Multiply by the height. 36 × 7 = 252, so V = 252π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 252π cm3
Check. Estimate with π ≈ 3: 3 × 36 × 7 = 756, which is close to 792. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.
- Write the volume formula with the numbers. We know V and r. The height h is unknown.150π = π × 52 × h
- Work out r2. 52 = 25, so 150π = 25π × h
- Divide both sides by π. This cancels the π on each side.150 = 25 × h
- Divide by the number in front of h. Undo the multiplication.h = 150 ÷ 25 = 6 cm
Check. π × 52 × 6 = 150π. ✓
Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.
Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.
- Write the volume formula with the numbers. We know V and r. The height h is unknown.252π = π × 62 × h
- Work out r2. 62 = 36, so 252π = 36π × h
- Divide both sides by π. This cancels the π on each side.252 = 36 × h
- Divide by the number in front of h. Undo the multiplication.h = 252 ÷ 36 = 7 cm
Check. π × 62 × 7 = 252π. ✓
Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.
Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.
- Write the volume formula with the numbers. We know V and r. The height h is unknown.96π = π × 42 × h
- Work out r2. 42 = 16, so 96π = 16π × h
- Divide both sides by π. This cancels the π on each side.96 = 16 × h
- Divide by the number in front of h. Undo the multiplication.h = 96 ÷ 16 = 6 cm
Check. π × 42 × 6 = 96π. ✓
Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.
Method. Use V = πr2h with π replaced by 22 ÷ 7. Cancel the 7 with the 3.5 to keep the arithmetic easy.
- Substitute. V = 22/7 × 3.5 × 3.5 × 10
- Use 3.5 = 7 ÷ 2 to cancel the 7. 22/7 × 3.5 = 22 × 0.5 = 11, because 3.5 ÷ 7 = 0.5.22/7 × 3.5 = 11
- Multiply the rest. Now multiply 11 by the other 3.5 and by 10.11 × 3.5 = 38.5, then 38.5 × 10 = 385 cm3
Check. 3.5 × 3.5 = 12.25, and 12.25 × 10 = 122.5; 122.5 × 22 ÷ 7 = 385. ✓
Watch out. Square the radius (3.52) rather than doubling it.
Method. Work out both volumes in terms of π and compare them.
- Volume of A. V = πr2h.π × 22 × 9 = π × 4 × 9 = 36π
- Volume of B. π × 32 × 4 = π × 9 × 4 = 36π
- Compare. Both volumes are 36π cm3, so they are equal. ✓
Check. Leaving the answer in terms of π makes it easy to compare. ✓
Method. The prism is half a cylinder: use half of πr2h.
- Volume of the whole cylinder. r = 6, h = 10.π × 62 × 10 = π × 36 × 10 = 360π
- Halve it. A semicircle is half a circle.360π ÷ 2 = 180π cm3
Check. Area of semicircle = ½ × π × 36 = 18π, and 18π × 10 = 180π. ✓
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Density = mass ÷ volume.
- Write down the formula. Density is mass shared out over each unit of volume.Density = mass ÷ volume
- Substitute. Mass is in g and volume in cm3, so the answer will be in g/cm3.Density = 450 ÷ 50
- Divide. 450 ÷ 50 = 9
- Write the unit. Density = 9 g/cm3
Check. Mass = density × volume = 9 × 50 = 450 g. ✓
Watch out. Divide the mass by the volume, not the other way round.
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Mass = density × volume.
- Write down the formula. Rearrange Density = mass ÷ volume by multiplying both sides by the volume.Mass = density × volume
- Substitute. Density is 2.7 g/cm3: every cm3 has a mass of 2.7 g.Mass = 2.7 × 200
- Multiply. 2.7 × 200 = 540
- Write the unit. Mass = 540 g
Check. 540 ÷ 200 = 2.7, the density we started with. ✓
Watch out. Check that the volume is in the same unit as in the density.
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Volume = mass ÷ density.
- Write down the formula. Rearrange Density = mass ÷ volume by swapping the volume and the density.Volume = mass ÷ density
- Substitute. Volume = 600 ÷ 8
- Divide. 600 ÷ 8 = 75
- Write the unit. Mass in g divided by g/cm3 leaves cm3.Volume = 75 cm3
Check. 8 × 75 = 600 g. ✓
Watch out. Divide the mass by the density to get the volume.
Method. Find the volume of the cuboid first, then divide the mass by the volume.
- Volume of the cuboid. length × width × height.10 × 5 = 50, then 50 × 4 = 200 cm3
- Density = mass ÷ volume. 160 ÷ 200 = 0.8
- Write the unit. Grams per cubic centimetre.0.8 g/cm3
Check. 0.8 × 200 = 160 g. ✓
Watch out. Divide by the volume, not by a length.
Method. Use mass = density × volume. The units are already kg and m3.
- Write the formula and substitute. Mass = 750 × 0.04
- Multiply. 0.04 = 4 ÷ 100, so find 750 × 4 = 3000 and divide by 100.3000 ÷ 100 = 30
- Write the unit. 30 kg
Check. 750 × 0.04 = 30. ✓
Method. Work out the volume in cm3 in terms of π, then divide by 1000 to change to litres.
- Volume in cm3. V = πr2h.π × 102 × 20 = π × 100 × 20 = 2000π cm3
- Change to litres. 1 litre = 1000 cm3.2000π ÷ 1000 = 2π litres
- Conclude. The capacity is 2π litres. ✓
Check. 2π is about 6.3 litres, which is sensible for a tank 20 cm tall and 20 cm across. ✓
Method. Put the known numbers into V = πr2h, then cancel the π and divide to find h.
- Write the volume formula with the numbers. We know V and r. The height h is unknown.900π = π × 102 × h
- Work out r2. 102 = 100, so 900π = 100π × h
- Divide both sides by π. This cancels the π on each side.900 = 100 × h
- Divide by the number in front of h. Undo the multiplication.h = 900 ÷ 100 = 9 cm
Check. π × 102 × 9 = 900π. ✓
Watch out. Divide by π as well as by r2. If the volume is given in terms of π, the π simply cancels.
Method. Check that he used the radius in πr2h.
- Spot the mistake. The formula needs the radius. Ali used the diameter, 10, instead of the radius, 5.The radius is 10 ÷ 2 = 5 cm.
- Redo the calculation. V = π × 52 × 6 = π × 25 × 6 = 150π cm3
Check. Ali’s answer is 4 times too big, which makes sense because 102 is 4 times 52. ✓
Watch out. Halve the diameter first.
Method. Look at the formula V = πr2h.
- Replace h by 2h. The 2 can be moved to the front.πr2 × 2h = 2 × πr2h = 2V
- Conclude. The volume doubles.
Check. A cylinder twice as tall holds twice as much. ✓
Method. Look at the formula V = πr2h. The radius is squared.
- Replace r by 2r. (2r)2 = 22r2 = 4r2.π × (2r)2 × h = π × 4r2 × h = 4V
- Conclude. The volume is multiplied by 4.
Check. Radius 1 gives 1 × π × h. Radius 2 gives 4 × π × h. ✓
Watch out. Doubling the radius does not double the volume because r is squared.
Method. Area of a circle = πr2.
- Substitute r = 3. Area = π × 32
- Square 3. 3 × 3 = 9.Area = 9π cm2
Check. The area of a circle of radius 3 is a bit more than 3 × 9 = 27 cm2. ✓
Method. A cylinder is a prism: volume = area of the end × height.
- Write the formula. The area of the end is given, so no radius is needed.Volume = 25π × 4
- Multiply the numbers. 25 × 4 = 100.Volume = 100π cm3
Check. Radius is 5 (since πr2 = 25π): π × 25 × 4 = 100π. ✓
Method. Think about what density measures: mass in every unit of volume.
- Mass in each unit of volume. If 3 cm3 has a mass of 24 g, each cm3 has 24 ÷ 3 = 8 g.divide the mass by the volume
- Write the formula. density = mass ÷ volume
Check. Units: g ÷ cm3 = g/cm3. ✓
Exercise 19G Cylinders and density (calculator)
Calculator allowedKey words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 42 × 10
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.42 = 4 × 4 = 16
- Multiply by the height. 16 × 10 = 160, so V = 160π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 42 × 10 = 502.6548…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.502.6548… = 503 cm3
Check. Estimate with π ≈ 3: 3 × 16 × 10 = 480, which is close to 503. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 2.52 × 7.2
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.2.52 = 2.5 × 2.5 = 6.25
- Multiply by the height. 6.25 × 7.2 = 45, so V = 45π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 2.52 × 7.2 = 141.3717…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.141.3717… = 141 cm3
Check. Estimate with π ≈ 3: 3 × 6.25 × 7.2 = 135, which is close to 141. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Find the radius. The formulas use the radius, which is half the diameter.r = 12 ÷ 2 = 6
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 62 × 15
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.62 = 6 × 6 = 36
- Multiply by the height. 36 × 15 = 540, so V = 540π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 62 × 15 = 1696.4600…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.1696.4600… = 1700 cm3
Check. Estimate with π ≈ 3: 3 × 36 × 15 = 1620, which is close to 1696. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 0.82 × 1.5
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.0.82 = 0.8 × 0.8 = 0.64
- Multiply by the height. 0.64 × 1.5 = 0.96, so V = 0.96π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 0.82 × 1.5 = 3.0159…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.3.0159… = 3.02 m3
Check. Estimate with π ≈ 3: 3 × 0.64 × 1.5 = 2.88, which is close to 3. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 6.42 × 3
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.6.42 = 6.4 × 6.4 = 40.96
- Multiply by the height. 40.96 × 3 = 122.88, so V = 122.88π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 6.42 × 3 = 386.0389…
- Round the answer. The question wants 1 decimal place. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.386.0389… = 386.0 cm3
Check. Estimate with π ≈ 3: 3 × 40.96 × 3 = 368.64, which is close to 386. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Find the radius. The formulas use the radius, which is half the diameter.r = 6.6 ÷ 2 = 3.3
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 3.32 × 11
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.3.32 = 3.3 × 3.3 = 10.89
- Multiply by the height. 10.89 × 11 = 119.79, so V = 119.79π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 3.32 × 11 = 376.3314…
- Round the answer. The question wants 1 decimal place. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.376.3314… = 376.3 cm3
Check. Estimate with π ≈ 3: 3 × 10.89 × 11 = 359.37, which is close to 376. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Method. Work out the volume in cm3, remember that 1 ml = 1 cm3, and compare with 330 ml.
- Find the radius. r = 6.6 ÷ 2 = 3.3 cm
- Work out the volume. Use V = πr2h with the π button.V = π × 3.32 × 11 = π × 119.79 = 376.3 cm3
- Change to ml and compare. 1 cm3 = 1 ml. This is the full-to-the-top capacity, and cans are not filled to the brim.376 ml is more than 330 ml.
- Write a conclusion. The label is sensible: the can holds about 376 ml when full, so 330 ml of drink leaves room at the top.
Check. Estimate with π ≈ 3: 3 × 119.79 = 359.37, close to 376.3. ✓
Watch out. Remember that 1 ml = 1 cm3.
Method. Find the volume in m3, then use 1 m3 = 1000 litres.
- Volume in m3. V = πr2h.V = π × 0.82 × 1.5 = π × 0.96 = 3.0159… m3
- Change to litres. 1 m3 = 1000 litres, so multiply by 1000.3.0159… × 1000 = 3015.93…
- Round to the nearest litre. 3016 litres
Check. Estimate: 3 × 0.64 × 1.5 × 1000 = 2880. ✓
Watch out. 1 m3 is 1000 litres.
Method. The pipe is a big cylinder with a smaller cylinder (the hole) removed. Subtract the volumes.
- Name the two cylinders. The outer cylinder has radius 5 cm and the hole has radius 4 cm. Both are 20 cm long.
- Volume of the outer cylinder. Volume of a cylinder = πr2h.π × 52 × 20 = π × 25 × 20 = 500π
- Volume of the inner cylinder (the hole). Volume of a cylinder = πr2h.π × 42 × 20 = π × 16 × 20 = 320π
- Subtract the parts. We take away the part that is missing, so subtract. Keep the π terms together.500π − 320π = 180π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.500π − 320π = 565.4867…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.565.4867… = 565 cm3
Check. Estimate with π ≈ 3: the total is about 540, close to the answer 565. ✓
Method. Work out both volumes in terms of π, compare, then subtract.
- Volume of A. V = πr2h.π × 32 × 10 = 90π
- Volume of B. π × 42 × 6 = 96π
- Compare. 96 is bigger than 90, so can B holds more.B > A
- Find the difference. Subtract the π terms first: 96π − 90π = 6π.6π = 18.8496… = 18.8 cm3
Check. A = 282.7 and B = 301.6; the difference is 18.8. ✓
Method. Find the volume of the cylinder, then use mass = density × volume, then change grams to kilograms.
- Volume of the rod. V = πr2h with r = 1.5 and h = 40.π × 1.52 × 40 = π × 90 = 282.743… cm3
- Mass in grams. Mass = density × volume.7.8 × 282.743… = 2205.40… g
- Change to kilograms. 1 kg = 1000 g, so divide by 1000.2205.40… ÷ 1000 = 2.2054… kg
- Round to 3 significant figures. 2.21 kg
Check. Estimate: 3 × 2.25 × 40 = 270 cm3; 270 × 8 = 2160 g = 2.16 kg. ✓
Method. Change the capacity to cm3, then rearrange V = πr2h to find h = V ÷ (πr2).
- Change litres to cm3. 1 litre = 1000 cm3.2 × 1000 = 2000 cm3
- Write the formula and substitute. 2000 = π × 82 × h = 64π × h
- Divide both sides by 64π. This leaves h on its own.h = 2000 ÷ (64 × π) = 9.9472…
- Round to 3 significant figures. 9.95 cm
Check. π × 64 × 9.95 = 2001, which is about 2000. ✓
Watch out. Divide by π and by r2; do not forget either.
Method. The trough is half a cylinder. Work in cm throughout, then change to litres.
- Change the length to cm. 1 m = 100 cm.2 m = 200 cm
- Volume of the whole cylinder. V = πr2h with r = 15 and h = 200.π × 152 × 200 = 45,000π = 141,371.67… cm3
- Halve it. The cross-section is a semicircle.141,371.67… ÷ 2 = 70,685.83… cm3
- Change to litres. 1 litre = 1000 cm3.70,685.83… ÷ 1000 = 70.6858…
- Round to 3 significant figures. 70.7 litres
Check. Estimate: 3 × 225 × 200 ÷ 2 = 67,500 cm3, about 68 litres. ✓
Method. Find both volumes, divide the cylinder by the cube, and change the fraction to a percentage.
- Volume of the cylinder. V = πr2h.π × 52 × 10 = 250π = 785.398… cm3
- Volume of the cube. side × side × side.10 × 10 × 10 = 1000 cm3
- Find the fraction as a percentage. Divide the cylinder by the cube and multiply by 100.785.398… ÷ 1000 × 100 = 78.540…
- Round to 3 significant figures. 78.5%
Check. The ratio of the volumes is π : 4, and π ÷ 4 = 0.785… = 78.5%. ✓
Method. Find the volume of the cylinder, then use density = mass ÷ volume.
- Volume of the cylinder. V = πr2h.π × 22 × 5 = 20π = 62.8319… cm3
- Divide the mass by the volume. Density = mass ÷ volume.420 ÷ 62.8319… = 6.6845…
- Round to 1 decimal place. 6.7 g/cm3
Check. Estimate: 420 ÷ 60 = 7, close to the answer. ✓
Method. Treat the water in the pipe as a cylinder of radius 2 cm and length 5 m = 500 cm.
- Change the length to cm. 1 m = 100 cm.5 m = 500 cm
- Volume in cm3. V = πr2h with r = 2 and h = 500.π × 22 × 500 = 2000π = 6283.185… cm3
- Change to litres. 1 litre = 1000 cm3.6283.185… ÷ 1000 = 6.2832…
- Round to 3 significant figures. 6.28 litres
Check. 2π is about 6.3. ✓
Surface area of cuboids, prisms and cylinders
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
The surface area of a solid is the total area of all of its faces. It tells you how much wrapping paper you need for a present, or how much paint covers a tank. Because it is an area, it is measured in square units such as cm2 or m2. Do not mix it up with volume, which is the space inside (cubic units).
The best way to find a surface area is to think of the net (Section 19.2). The net shows every face flat, so you can work out each area and add.
Cuboid. There are six rectangles in three matching pairs. Surface area = 2lw + 2lh + 2wh. For a cube with edge a all six faces are the same: surface area = 6a2.
Prisms. A prism has two identical end faces (the cross-section) and a rectangle along each side. Each rectangle is one edge of the cross-section multiplied by the length of the prism. So
surface area = 2 × (area of cross-section) + (perimeter of cross-section) × length
The second part works because all the rectangles have the same length, so together they make one big rectangle: the perimeter by the length.
Cylinders. The net of a cylinder is two circles and a rectangle. The rectangle wraps round the side, so its width is the circumference 2πr and its height is h.
surface area of a cylinder = 2πr2 + 2πrh
The first part is the two circles, the second part is the curved surface. If the cylinder has no lid (an open can) there is only one circle, so the surface area is πr2 + 2πrh.
Key ideas
- Surface area = total area of all the faces, in square units.
- Cuboid: 2lw + 2lh + 2wh. Cube: 6a2.
- Prism: 2 × (cross-section area) + (perimeter of cross-section) × length.
- Cylinder: 2πr2 + 2πrh (curved surface 2πrh). An open cylinder has only one circle.
- Use the net to check that you have included every face.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.5 × 4 = 20 (and there are 2 of them, so 40)
- Front and back. Each is length × height.5 × 3 = 15 (2 of them: 30)
- Left and right. Each is width × height.4 × 3 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.40 + 30 + 24 = 94
- Write the unit. Area, so square units.Surface area = 94 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (20 + 15 + 12) = 2 × 47 = 94. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. It is a right-angled triangle, so use the two shorter sides as base and height: ½ × base × height.½ × 3 × 4 = 6 cm2
- Double it for the two ends. The two end faces are identical.2 × 6 = 12 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the side of length 3: 3 × 10 = 30
Rectangle on the side of length 4: 4 × 10 = 40
Rectangle on the side of length 5: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.12 + 30 + 40 + 50 = 132
- Write the unit. Area, so square units.Surface area = 132 cm2
Check. The rectangles together make (perimeter of an end) × length = 12 × 10 = 120. Then 12 + 120 = 132. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (L-shapes) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Double it for the two ends. The two end faces are identical.2 × 28 = 56 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Bottom edge: 8 × 10 = 80
Right edge: 2 × 10 = 20
Inner horizontal edge: 5 × 10 = 50
Inner vertical edge: 4 × 10 = 40
Top edge: 3 × 10 = 30
Left edge: 6 × 10 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.56 + 80 + 20 + 50 + 40 + 30 + 60 = 336
- Write the unit. Area, so square units.Surface area = 336 cm2
Check. The rectangles together make (perimeter of an end) × length = 28 × 10 = 280. Then 56 + 280 = 336. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 32 = 18π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 3 × 10 = 60π
- Add them together. Total surface area = circles + curved surface.18π + 60π = 78π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 78π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. Work out the area of the five faces (base and four sides, no lid) and multiply by the cost per square metre.
- Area of the base. length × width.2 × 1.5 = 3 m2
- Area of the two long sides. Each is length × height.2 × 1 = 2 m2, so 2 of them: 4 m2
- Area of the two short sides. Each is width × height.1.5 × 1 = 1.5 m2, so 2 of them: 3 m2
- Add to get the area to paint. There is no lid so we do not count one.3 + 4 + 3 = 10 m2
- Multiply by the cost. £4 for each m2.10 × 4 = £40
Check. A closed tank would be 13 m2; removing the lid (3 m2) leaves 10 m2. ✓
Watch out. Check whether the question includes the top. An open tank has 5 faces.
Watch out
Do not forget faces! Use the net to count them. A cuboid has six faces, a triangular prism has five, and a closed cylinder has two circles and one curved surface. Give the answer in square units, and do not confuse surface area with volume.Exercise 19H Surface area (non-calculator)
Non-calculatorKey words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.5 × 4 = 20 (and there are 2 of them, so 40)
- Front and back. Each is length × height.5 × 3 = 15 (2 of them: 30)
- Left and right. Each is width × height.4 × 3 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.40 + 30 + 24 = 94
- Write the unit. Area, so square units.Surface area = 94 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (20 + 15 + 12) = 2 × 47 = 94. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.6 × 6 = 36 (and there are 2 of them, so 72)
- Front and back. Each is length × height.6 × 2 = 12 (2 of them: 24)
- Left and right. Each is width × height.6 × 2 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.72 + 24 + 24 = 120
- Write the unit. Area, so square units.Surface area = 120 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (36 + 12 + 12) = 2 × 60 = 120. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.10 × 3 = 30 (and there are 2 of them, so 60)
- Front and back. Each is length × height.10 × 2 = 20 (2 of them: 40)
- Left and right. Each is width × height.3 × 2 = 6 (2 of them: 12)
- Add the three pairs. The surface area is every face added together.60 + 40 + 12 = 112
- Write the unit. Area, so square units.Surface area = 112 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (30 + 20 + 6) = 2 × 56 = 112. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.4 × 3 = 12 (and there are 2 of them, so 24)
- Front and back. Each is length × height.4 × 2.5 = 10 (2 of them: 20)
- Left and right. Each is width × height.3 × 2.5 = 7.5 (2 of them: 15)
- Add the three pairs. The surface area is every face added together.24 + 20 + 15 = 59
- Write the unit. Area, so square units.Surface area = 59 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (12 + 10 + 7.5) = 2 × 29.5 = 59. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.
- Area of one face. Each face is a square with side equal to the edge length.4 × 4 = 16
- Multiply by the number of faces. A cube has 6 faces.6 × 16 = 96
- Write the unit. Surface area = 96 cm2
Check. Formula 6a2 = 6 × 42 = 6 × 16 = 96. ✓
Watch out. Do not confuse with the volume: volume cubes the side (a3), surface area squares it (6a2).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.
- Area of one face. Each face is a square with side equal to the edge length.7 × 7 = 49
- Multiply by the number of faces. A cube has 6 faces.6 × 49 = 294
- Write the unit. Surface area = 294 cm2
Check. Formula 6a2 = 6 × 72 = 6 × 49 = 294. ✓
Watch out. Do not confuse with the volume: volume cubes the side (a3), surface area squares it (6a2).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.
- Area of one face. Each face is a square with side equal to the edge length.10 × 10 = 100
- Multiply by the number of faces. A cube has 6 faces.6 × 100 = 600
- Write the unit. Surface area = 600 mm2
Check. Formula 6a2 = 6 × 102 = 6 × 100 = 600. ✓
Watch out. Do not confuse with the volume: volume cubes the side (a3), surface area squares it (6a2).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the five faces. A box with no lid has 5 faces: the base and four sides. The top is missing, so we do not count it.
- The base. Length × width, one face only.20 × 15 = 300
- The long sides (front and back). Each is length × height and there are 2.20 × 10 = 200, so 2 of them: 400
- The short sides (left and right). Each is width × height and there are 2.15 × 10 = 150, so 2 of them: 300
- Add them. 300 + 400 + 300 = 1000
- Write the unit. Area, so square units.Surface area = 1000 cm2
Check. A closed box would be 1300; taking away the lid (300) leaves 1000. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. It is a right-angled triangle, so use the two shorter sides as base and height: ½ × base × height.½ × 3 × 4 = 6 cm2
- Double it for the two ends. The two end faces are identical.2 × 6 = 12 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the side of length 3: 3 × 10 = 30
Rectangle on the side of length 4: 4 × 10 = 40
Rectangle on the side of length 5: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.12 + 30 + 40 + 50 = 132
- Write the unit. Area, so square units.Surface area = 132 cm2
Check. The rectangles together make (perimeter of an end) × length = 12 × 10 = 120. Then 12 + 120 = 132. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (12 cm).
- Area of one triangular end. Use the perpendicular height of the triangle, not the sloping side.½ × 6 × 4 = 12 cm2
- Double it for the two ends. The two end faces are identical.2 × 12 = 24 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle under the base: 6 × 12 = 72
First sloping rectangle: 5 × 12 = 60
Second sloping rectangle: 5 × 12 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.24 + 72 + 60 + 60 = 216
- Write the unit. Area, so square units.Surface area = 216 cm2
Check. The rectangles together make (perimeter of an end) × length = 16 × 12 = 192. Then 24 + 192 = 216. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (trapeziums) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one trapezium end. Area of a trapezium = ½ × (sum of the parallel sides) × height.½ × (4 + 8) × 3 = 18 cm2
- Double it for the two ends. The two end faces are identical.2 × 18 = 36 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the top side: 4 × 10 = 40
Rectangle on the bottom side: 8 × 10 = 80
First sloping rectangle: 5 × 10 = 50
Second sloping rectangle: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.36 + 40 + 80 + 50 + 50 = 256
- Write the unit. Area, so square units.Surface area = 256 cm2
Check. The rectangles together make (perimeter of an end) × length = 22 × 10 = 220. Then 36 + 220 = 256. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (L-shapes) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Double it for the two ends. The two end faces are identical.2 × 28 = 56 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Bottom edge: 8 × 10 = 80
Right edge: 2 × 10 = 20
Inner horizontal edge: 5 × 10 = 50
Inner vertical edge: 4 × 10 = 40
Top edge: 3 × 10 = 30
Left edge: 6 × 10 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.56 + 80 + 20 + 50 + 40 + 30 + 60 = 336
- Write the unit. Area, so square units.Surface area = 336 cm2
Check. The rectangles together make (perimeter of an end) × length = 28 × 10 = 280. Then 56 + 280 = 336. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 32 = 18π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 3 × 10 = 60π
- Add them together. Total surface area = circles + curved surface.18π + 60π = 78π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 78π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Find the radius. The formulas use the radius, which is half the diameter.r = 8 ÷ 2 = 4
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 42 = 32π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 4 × 5 = 40π
- Add them together. Total surface area = circles + curved surface.32π + 40π = 72π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 72π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 4 × 9 = 72π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 52 = 50π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 5 × 2 = 20π
- Add them together. Total surface area = circles + curved surface.50π + 20π = 70π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 70π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 5 × 7 = 70π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 4 × 10 = 80π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Curved surface area = 80π cm2
Check. Circumference = 2πr = 8π; times the height 10 = 80π. ✓
Watch out. The curved surface does not include the circular ends.
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 3 × 7 = 42π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Curved surface area = 42π cm2
Check. Circumference = 2πr = 6π; times the height 7 = 42π. ✓
Watch out. The curved surface does not include the circular ends.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of an open cylinder = one circle + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into one circle (the base) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the base circle. Area of a circle = πr2. Only one circle because there is no lid.π × 52 = 25π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 5 × 12 = 120π
- Add them together. Total surface area = circles + curved surface.25π + 120π = 145π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 145π cm2
Check. Using πr(r + 2h) = π × 5 × 29 = 145π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. Find the surface area of the box and the area of one sheet, then divide and round up.
- Surface area of the box. Three pairs of faces: 30 × 20, 30 × 10, 20 × 10.2 × (600 + 300 + 200) = 2 × 1100 = 2200 cm2
- Area of one sheet. 50 × 40.2000 cm2
- Divide. See how many sheets are needed.2200 ÷ 2000 = 1.1
- Round up. 1.1 sheets means more than one sheet is needed, so 2 whole sheets.2 sheets
Check. 1 sheet gives 2000 cm2, which is less than 2200. 2 sheets give 4000, which is enough. ✓
Watch out. Round up when you need whole sheets.
Method. Find the area of the five faces and multiply by the cost per m2.
- Area of the five faces. Use the open-top cuboid surface area: base + 2 long sides + 2 short sides.3 + 2 × 2 + 2 × 1.5 = 3 + 4 + 3 = 10 m2
- Multiply by the cost. £4 for each m2.10 × 4 = £40
Check. 10 m2 at £4 is £40. ✓
Method. Work backwards: divide the total by 6 to get one face, then find the square root.
- Area of one face. A cube has 6 identical faces.216 ÷ 6 = 36 cm2
- Find the edge. Each face is a square, so edge × edge = 36. Since 6 × 6 = 36 the edge is 6 cm.Edge = 6 cm
Check. 6 × 62 = 6 × 36 = 216. ✓
Watch out. Square root (not divide by 2) to get the edge from the area of a face.
Method. Work out the volume and surface area of both solids and compare.
- Volume of the cube. edge3.6 × 6 × 6 = 216 cm3
- Volume of the cuboid. length × width × height.9 × 6 = 54, then 54 × 4 = 216 cm3
- Surface area of the cube. 6 faces of 36.6 × 36 = 216 cm2
- Surface area of the cuboid. 2(lw + lh + wh).2 × (54 + 36 + 24) = 2 × 114 = 228 cm2
- Compare. Same volume. The cube has the smaller surface area: 216 < 228.
Check. A cube is the most compact cuboid, so its surface area is the smallest for that volume. ✓
Method. Work out the curved surface and the two circles separately and compare with 60π.
- Curved surface. 2πrh with r = 3 and h = 10.2 × π × 3 × 10 = 60π
- Tom’s mistake. His answer is only the curved surface. He left out the two circular ends.Two circles: 2 × π × 32 = 18π
- Add all the parts. 60π + 18π = 78π cm2
Check. 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Method. Work out the surface area before, then of both halves after, and subtract.
- Surface area of the original. 2(lw + lh + wh).2 × (24 + 18 + 12) = 2 × 54 = 108 cm2
- Surface area of one half. Use 6 by 4 by 1.5.2 × (24 + 9 + 6) = 2 × 39 = 78 cm2
- Both halves. There are two identical halves.2 × 78 = 156 cm2
- Find the increase. Subtract.156 − 108 = 48 cm2. ✓
Check. The cut makes two new faces of 6 × 4 = 24 each, and 2 × 24 = 48. ✓
Method. Find the surface area of the cuboid and of the two cubes and compare.
- Surface area of the cuboid. 2(lw + lh + wh) with 4, 2 and 2.2 × (8 + 8 + 4) = 2 × 20 = 40 cm2
- Surface area of two separate cubes. Each cube is 6 × 2 × 2 = 24 cm2.2 × 24 = 48 cm2
- Explain the difference. The faces that touch are no longer on the outside. Each is 2 × 2 = 4 cm2 and there are 2 of them.48 − 40 = 8 = 2 × 4 ✓
Check. The hidden faces total 8 cm2. ✓
Method. Each face of a cube is a square.
- Write the area formula for a square. Area = side × side.Area = 3 × 3
- Multiply. 9 cm2
Check. A 3 by 3 grid has 9 squares. ✓
Method. A cuboid has 6 faces in 3 matching pairs. Work out one face from each pair.
- Top and bottom. 4 × 3 = 12, and 2 of them.12 + 12 = 24
- Front and back. 4 × 2 = 8, and 2 of them.8 + 8 = 16
- Left and right. 3 × 2 = 6, and 2 of them.6 + 6 = 12
- Add them. 24 + 16 + 12 = 52 cm2
Check. 2(12 + 8 + 6) = 2 × 26 = 52. ✓
Exercise 19I Surface area of cylinders (calculator)
Calculator allowedKey words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 42 = 32π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 4 × 10 = 80π
- Add them together. Total surface area = circles + curved surface.32π + 80π = 112π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.112π = 351.8584…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.351.8584… = 352 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 4 × 14 = 112π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 2.52 = 12.5π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 2.5 × 7.2 = 36π
- Add them together. Total surface area = circles + curved surface.12.5π + 36π = 48.5π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.48.5π = 152.3672…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.152.3672… = 152 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 2.5 × 9.7 = 48.5π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Find the radius. The formulas use the radius, which is half the diameter.r = 12 ÷ 2 = 6
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 62 = 72π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 6 × 15 = 180π
- Add them together. Total surface area = circles + curved surface.72π + 180π = 252π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.252π = 791.6813…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.791.6813… = 792 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 6 × 21 = 252π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of an open cylinder = one circle + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into one circle (the base) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the base circle. Area of a circle = πr2. Only one circle because there is no lid.π × 62 = 36π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 6 × 14 = 168π
- Add them together. Total surface area = circles + curved surface.36π + 168π = 204π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.204π = 640.8849…
- Round the answer. The question wants 1 decimal place. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.640.8849… = 640.9 cm2
Check. Using πr(r + 2h) = π × 6 × 34 = 204π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 3.7 × 10.8 = 79.92π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.2 × π × 3.7 × 10.8 = 251.0761…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.251.0761… = 251 cm2
Check. Circumference = 2πr = 7.4π; times the height 10.8 = 79.92π. ✓
Watch out. The curved surface does not include the circular ends.
Method. Find the surface area of the base and curved surface, then divide by the area one litre covers. Round up because you cannot buy part of a litre.
- Area of the base circle. Only one circle because the tank is open.π × 0.82 = 0.64π
- Area of the curved surface. 2πrh.2 × π × 0.8 × 1.5 = 2.4π
- Add and use the calculator. 0.64π + 2.4π = 3.04π.3.04π = 9.550… m2
- Divide by 4 m2 per litre. This gives the litres needed.9.550… ÷ 4 = 2.388…
- Round up. Part of a litre must be a whole extra litre.3 litres
Check. 2 litres would cover only 8 m2, which is less than the area. ✓
Watch out. Round up when you need to buy whole units.
Method. Work out both areas in terms of π and write them as a ratio. The π cancels.
- Curved surface area. 2πrh.2 × π × 3 × 8 = 48π
- Total surface area. 2πr2 + 2πrh.2 × π × 9 + 48π = 18π + 48π = 66π
- Write the ratio. The π is in both so it cancels.48 : 66
- Simplify. Divide both parts by 6.48 : 66 = 8 : 11
Check. 8 × 6 = 48 and 11 × 6 = 66. ✓
Chapter summary
| Skill | Method |
|---|---|
| Properties of solids | Face = flat surface, edge = where two faces meet, vertex = corner. Euler: V − E + F = 2. n-sided prism: n + 2 faces, 3n edges, 2n vertices. n-sided pyramid: n + 1 faces, 2n edges, n + 1 vertices |
| Nets | A net folds to make the solid. Cube: 6 squares (11 different nets). Cylinder: rectangle 2πr by h and two circles. Cone (H): sector of radius l and angle 360r ÷ l |
| Plans and elevations | Plan = from above, front elevation = from the front, side elevation = from the side (front on the left). Use the tallest stack in each column or row |
| Volume of a cuboid | length × width × height. Cube: side3 |
| Units of volume | 1 ml = 1 cm3. 1 litre = 1000 cm3. 1 m3 = 1000 litres = 1,000,000 cm3 |
| Volume of a prism | area of cross-section × length (use the perpendicular height) |
| Cylinder | V = πr2h. Total surface area = 2πr2 + 2πrh. Curved surface = 2πrh |
| Density | density = mass ÷ volume; mass = density × volume; volume = mass ÷ density |
| Surface area of a cuboid or prism | Cuboid: 2lw + 2lh + 2wh. Prism: 2 × (cross-section area) + (perimeter of the cross-section) × length |
| Pyramid and cone (H) | Pyramid V = ⅓ × base area × vertical height. Cone V = ⅓πr2h, curved surface πrl, total πrl + πr2. l2 = r2 + h2 |
| Sphere and hemisphere (H) | Sphere V = &frac43;πr3, surface 4πr2. Hemisphere V = ⅔πr3, curved 2πr2, total 3πr2 |
| Composite solids and frustums (H) | Split into standard solids; add volumes, subtract holes; count only outside surfaces. Frustum = large cone − small cone (similar triangles) |
| Similar solids (H) | Length scale factor k: area × k2, volume (and mass) × k3 |
Can you…?
- I can name solids and count their faces, edges and vertices, and use Euler’s formula.
- I can recognise nets of cubes, cuboids, prisms, pyramids and cylinders.
- I can draw the plan and elevations of a solid made of cubes and of standard solids.
- I can work out the volume of a cuboid, prism and cylinder, and convert between cm3, litres and m3.
- I can use density = mass ÷ volume.
- I can work out the surface area of a cuboid, prism and cylinder.
- Higher: I can work out the volume and surface area of pyramids, cones, spheres and hemispheres.
- Higher: I can solve composite-solid, frustum, working-backwards and similar-solid problems.
Key terms
- Solid (3D shape)
- A shape with length, width and height.
- Face
- A flat surface of a solid.
- Edge
- A line where two faces of a solid meet.
- Vertex
- A corner of a solid (plural: vertices).
- Prism
- A solid with the same cross-section all along its length.
- Cross-section
- The face you see when you slice straight through a prism.
- Pyramid
- A solid with a polygon base and triangular faces meeting at an apex.
- Net
- A flat pattern that folds up to make a solid.
- Plan
- The view of a solid from above.
- Elevation
- A view of a solid from the front or the side.
- Volume
- The amount of space inside a solid, in cubic units.
- Capacity
- The amount a container can hold, in litres or millilitres.
- Surface area
- The total area of all the faces of a solid.
- Density
- The mass of a material in each unit of volume.
- Slant height
- The length along a sloping surface of a cone or pyramid.
- Frustum
- A cone with the top cut off by a plane parallel to the base.
- Similar solids
- Solids that are enlargements of each other, with all lengths in the same ratio.
Exercise 19Z Check-up (non-calculator)
Non-calculatorMethod. Compare the numbers of faces, edges and vertices with the table at the start of this section.
- Write the counts. Faces = 5, edges = 9, vertices = 6
- Find the matching solid in the table. The numbers of faces, edges and vertices are different for each solid, so only one fits.5, 9, 6 matches the triangular prism.
Check. Euler: 6 − 9 + 5 = 2. ✓
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 6 × 5 × 4
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.6 × 5 = 30
- Multiply by the height. This counts all the layers.30 × 4 = 120
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 120 cm3
Check. Multiply in a different order: 5 × 4 = 20, then 20 × 6 = 120. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Method. The rectangle in the net of a cylinder is as wide as the circumference of the circular end.
- Think about how the rectangle wraps round. When the net is folded up, the long edge of the rectangle goes right round the circular end once. So it is as long as the circumference of the circle.
- Use the circumference formula. Circumference = 2πr (or πd).C = 2 × π × 5 = 10π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Width = 10π cm
Check. Circumference is a bit more than 3 × the diameter: 3 × 10 = 30. ✓
Watch out. Use the circumference (2πr) for the width of the rectangle, not the area of the circle.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.5 × 3 = 15 (and there are 2 of them, so 30)
- Front and back. Each is length × height.5 × 2 = 10 (2 of them: 20)
- Left and right. Each is width × height.3 × 2 = 6 (2 of them: 12)
- Add the three pairs. The surface area is every face added together.30 + 20 + 12 = 62
- Write the unit. Area, so square units.Surface area = 62 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (15 + 10 + 6) = 2 × 31 = 62. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. Capacity is how much a container can hold, measured in litres (l) and millilitres (ml). Volume is the space a solid takes up.
Method. Change the volume to cm3 first, using the fact that links the unit to cm3, then change to the unit you want.
- Decide which conversion fact we need. Every volume unit can be linked to cm3.1 litre = 1000 ml = 1000 cm3
- Change litres into cm3. 1 litre = 1000 ml = 1000 cm3.2.5 litres = 2.5 × 1000 = 2500 cm3
- Write the answer. Check that the unit matches what the question asked for.2.5 litres = 2500 cm3
Check. Go back the other way: 2500 cm3 gives 2.5 litres. ✓
Watch out. Volume units are not the same as length units: 1 m3 is 1,000,000 cm3, not 100 cm3.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: triangle. Length = 10 cm
- Work out the area of the triangular cross-section. Area of a triangle = ½ × base × perpendicular height.A = ½ × 8 × 3 = 24 ÷ 2 = 12 cm2
- Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 12 × 10 = 120
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 120 cm3
Check. A triangular prism is half of a cuboid 8 × 3 × 10 = 240, and half of that is 120. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 22 × 5
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.22 = 2 × 2 = 4
- Multiply by the height. 4 × 5 = 20, so V = 20π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.V = 20π cm3
Check. Estimate with π ≈ 3: 3 × 4 × 5 = 60, which is close to 63. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 32 = 18π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 3 × 4 = 24π
- Add them together. Total surface area = circles + curved surface.18π + 24π = 42π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Surface area = 42π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 3 × 7 = 42π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Key words. Density tells you how much mass is packed into each unit of volume (for example grams in every cm3). Dense materials like lead feel heavy for their size.
Method. Density = mass ÷ volume.
- Write down the formula. Density is mass shared out over each unit of volume.Density = mass ÷ volume
- Substitute. Mass is in g and volume in cm3, so the answer will be in g/cm3.Density = 600 ÷ 200
- Divide. 600 ÷ 200 = 3
- Write the unit. Density = 3 g/cm3
Check. Mass = density × volume = 3 × 200 = 600 g. ✓
Watch out. Divide the mass by the volume, not the other way round.
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A cone has a point at the top and a circle at the bottom.
- Say what you see. From the front the sloping sides meet at the point.The view is a triangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Total 30 marks. Allow about 30 minutes.
Exercise 19X Exam-style questions
Non-calculatorKey words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.8 × 5 = 40 (and there are 2 of them, so 80)
- Front and back. Each is length × height.8 × 3 = 24 (2 of them: 48)
- Left and right. Each is width × height.5 × 3 = 15 (2 of them: 30)
- Add the three pairs. The surface area is every face added together.80 + 48 + 30 = 158
- Write the unit. Area, so square units.Surface area = 158 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (40 + 24 + 15) = 2 × 79 = 158. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. A prism is a 3D shape with the same cross-section along its whole length. The cross-section is the face you would see if you sliced straight through it.
Method. For a prism: Volume = area of the cross-section × length.
- Identify the cross-section and the length. A prism has the same cross-section all the way along. The length of the prism is the distance between its two identical end faces.Cross-section: trapezium. Length = 12 cm
- Work out the area of the trapezium cross-section. Area of a trapezium = ½ × (sum of the parallel sides) × the distance between them.A = ½ × (4 + 8) × 3
A = ½ × 12 × 3 = 36 ÷ 2 = 18 cm2 - Multiply the area by the length. Volume of a prism = area of cross-section × length. Think of the cross-section as a slice, and the prism as that slice repeated along the length.V = 18 × 12 = 216
- Write the answer with its unit. Cubic units, because the volume is area (cm2) × length (cm).V = 216 cm3
Check. Estimate: the volume must be between 144 and 288 (the volumes of the prisms using only the shorter or only the longer parallel side), and 216 is in that range. ✓
Watch out. Use the perpendicular height of the triangle or trapezium, not a sloping side. The prism length is the distance between the two ends.
Key words. The radius (r) goes from the centre to the edge of a circle; the diameter is twice the radius. In terms of π means leave π as a symbol in the answer.
Method. Volume of a cylinder = area of the circular end × height = πr2h.
- Find the radius. The formulas use the radius, which is half the diameter.r = 10 ÷ 2 = 5
- Write down the formula. A cylinder is a circular prism: the cross-section is a circle (area πr2) and we multiply by the height.V = πr2h
- Substitute the radius and the height. V = π × 52 × 6
- Square the radius first. Powers come before multiplying (BIDMAS). r2 means r × r.52 = 5 × 5 = 25
- Multiply by the height. 25 × 6 = 150, so V = 150π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.π × 52 × 6 = 471.2389…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.471.2389… = 471 cm3
Check. Estimate with π ≈ 3: 3 × 25 × 6 = 450, which is close to 471. ✓
Watch out. Square only the radius, not π or the height. If you are given the diameter, halve it first.
Method. Find the volume in cm3, then divide by 1000 to change cm3 into litres.
- Work out the volume of the cuboid. Volume = length × width × height, all in cm.60 × 30 = 1800
1800 × 40 = 72,000 cm3 - Change cm3 to litres. 1 litre = 1000 cm3, so divide by 1000.72,000 ÷ 1000 = 72
- Write the answer. Capacity = 72 litres
Check. 72 litres × 1000 = 72,000 cm3. ✓
Watch out. 1 litre is 1000 cm3. Do not divide by 100.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. It is a right-angled triangle, so use the two shorter sides as base and height: ½ × base × height.½ × 3 × 4 = 6 cm2
- Double it for the two ends. The two end faces are identical.2 × 6 = 12 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the side of length 3: 3 × 10 = 30
Rectangle on the side of length 4: 4 × 10 = 40
Rectangle on the side of length 5: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.12 + 30 + 40 + 50 = 132
- Write the unit. Area, so square units.Surface area = 132 cm2
Check. The rectangles together make (perimeter of an end) × length = 12 × 10 = 120. Then 12 + 120 = 132. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Method. Find the volume, change the mass to grams, then divide the mass by the volume.
- Volume of the block. length × width × height.10 × 5 × 4 = 200 cm3
- Change the mass to grams. 1 kg = 1000 g.1.6 × 1000 = 1600 g
- Divide mass by volume. Density = mass ÷ volume.1600 ÷ 200 = 8 g/cm3
Check. 8 × 200 = 1600 g. ✓
Watch out. Marks: 1 for the volume, 1 for changing to grams, 1 for the density with its unit.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 1, 2, 3 → tallest = 3
Column 2 (from the left): stacks 0, 1, 2 → tallest = 2
Column 3 (from the left): stacks 0, 0, 1 → tallest = 1 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 3, 2, 1
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 6 squares: 3 + 2 + 1 = 6. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area of an open cylinder = one circle + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into one circle (the base) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the base circle. Area of a circle = πr2. Only one circle because there is no lid.π × 62 = 36π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 6 × 14 = 168π
- Add them together. Total surface area = circles + curved surface.36π + 168π = 204π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.204π = 640.8849…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.640.8849… = 641 cm2
Check. Using πr(r + 2h) = π × 6 × 34 = 204π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. Find the volume of the tent, then divide by the air each person needs.
- Area of the triangular end. ½ × base × height.½ × 4 × 3 = 6 m2
- Volume of the tent. Area of cross-section × length.6 × 5 = 30 m3
- Divide by the air for each person. 30 ÷ 2.5 = 12
- State the answer. The number is whole so no rounding is needed.12 people
Check. 12 × 2.5 = 30. ✓
Watch out. Marks: 1 for the area, 1 for the volume, 1 for dividing by 2.5, 1 for the answer.