GCSE Maths: Properties of 3D solids (Foundation)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Properties of 3D solids
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. ✓