GCSE Maths Volume, Surface Area and 3D Shapes (Higher): check-up and exam-style questions

Method. Compare the numbers of faces, edges and vertices with the table at the start of this section.

1. Write the counts.
   Faces = 5, edges = 9, vertices = 6
2. 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.

1. 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
2. Substitute the lengths. All three lengths are in cm, so the units match.
   V = 6 × 5 × 4
3. Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.
   6 × 5 = 30
4. Multiply by the height. This counts all the layers.
   30 × 4 = 120
5. 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.

1. 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.
2. Use the circumference formula. Circumference = 2πr (or πd).
   C = 2 × π × 5 = 10π
3. 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.

1. 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.
2. Top and bottom. Each is length × width.
   5 × 3 = 15  (and there are 2 of them, so 30)
3. Front and back. Each is length × height.
   5 × 2 = 10  (2 of them: 20)
4. Left and right. Each is width × height.
   3 × 2 = 6  (2 of them: 12)
5. Add the three pairs. The surface area is every face added together.
   30 + 20 + 12 = 62
6. 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.

1. Decide which conversion fact we need. Every volume unit can be linked to cm³.
   1 litre = 1000 ml = 1000 cm³
2. Change litres into cm³. 1 litre = 1000 ml = 1000 cm³.
   2.5 litres = 2.5 × 1000 = 2500 cm³
3. 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.

1. 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
2. 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
3. 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
4. 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.

1. 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
2. Substitute the radius and the height.
   V = π × 2² × 5
3. Square the radius first. Powers come before multiplying (BIDMAS). r² means r × r.
   2² = 2 × 2 = 4
4. Multiply by the height.
   4 × 5 = 20, so V = 20π
5. 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).

1. 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.
2. Area of the two circles. Area of a circle = πr². There are two of them.
   2 × π × 3² = 18π
3. Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.
   2 × π × 3 × 4 = 24π
4. Add them together. Total surface area = circles + curved surface.
   18π + 24π = 42π
5. 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.

1. Write down the formula. Density is mass shared out over each unit of volume.
   Density = mass ÷ volume
2. Substitute. Mass is in g and volume in cm³, so the answer will be in g/cm³.
   Density = 600 ÷ 200
3. Divide.
   600 ÷ 200 = 3
4. 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.

1. Picture the solid. A cone has a point at the top and a circle at the bottom.
2. 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. ✓

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.

1. 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.
2. Top and bottom. Each is length × width.
   8 × 5 = 40  (and there are 2 of them, so 80)
3. Front and back. Each is length × height.
   8 × 3 = 24  (2 of them: 48)
4. Left and right. Each is width × height.
   5 × 3 = 15  (2 of them: 30)
5. Add the three pairs. The surface area is every face added together.
   80 + 48 + 30 = 158
6. 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.

1. 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
2. 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
3. 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
4. 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.

1. Find the radius. The formulas use the radius, which is half the diameter.
   r = 10 ÷ 2 = 5
2. 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
3. Substitute the radius and the height.
   V = π × 5² × 6
4. Square the radius first. Powers come before multiplying (BIDMAS). r² means r × r.
   5² = 5 × 5 = 25
5. Multiply by the height.
   25 × 6 = 150, so V = 150π
6. 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…
7. 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.

1. Work out the volume of the cuboid. Volume = length × width × height, all in cm.
   60 × 30 = 1800
   1800 × 40 = 72,000 cm³
2. Change cm³ to litres. 1 litre = 1000 cm³, so divide by 1000.
   72,000 ÷ 1000 = 72
3. 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.

1. 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).
2. 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
3. Double it for the two ends. The two end faces are identical.
   2 × 6 = 12 cm²
4. 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
5. Add all the faces together. Surface area = the two ends + all the rectangles.
   12 + 30 + 40 + 50 = 132
6. 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.

1. Volume of the block. length × width × height.
   10 × 5 × 4 = 200 cm³
2. Change the mass to grams. 1 kg = 1000 g.
   1.6 × 1000 = 1600 g
3. 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.

1. 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.
2. 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
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: 3, 2, 1
4. 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).

1. 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.
2. Area of the base circle. Area of a circle = πr². Only one circle because there is no lid.
   π × 6² = 36π
3. Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.
   2 × π × 6 × 14 = 168π
4. Add them together. Total surface area = circles + curved surface.
   36π + 168π = 204π
5. 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…
6. 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.

1. Area of the triangular end. ½ × base × height.
   ½ × 4 × 3 = 6 m²
2. Volume of the tent. Area of cross-section × length.
   6 × 5 = 30 m³
3. Divide by the air for each person.
   30 ÷ 2.5 = 12
4. 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.

Key words. The slant height (l) of a cone is the distance from the point (apex) down the sloping side to the edge of the base. The vertical height (h) goes straight down from the apex to the centre of the base. They are different lengths.

Method. Volume of a cone = one third of πr²h, where h is the vertical height.

1. Write down the formula. A cone is one third of the cylinder with the same base and height.
   V = ⅓πr²h
2. Substitute.
   V = ⅓ × π × 6² × 8
3. Work out r² and multiply by h. Powers first, then multiply.
   6² = 36
   36 × 8 = 288
4. Take one third. Divide by 3.
   288 ÷ 3 = 96, so V = 96π
5. 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.
   96π cm³

Check. The cylinder with the same r and h has volume 288π; the cone is one third of it: 96π. ✓

Watch out. Use the vertical height h in V = ⅓πr²h. If you are given the slant height, use Pythagoras to find h first.

Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.

Method. Total surface area of a solid hemisphere = curved surface (2πr²) + flat circle (πr²) = 3πr².

1. Curved surface. Half of the sphere surface area 4πr².
   2πr² = 2 × π × 6² = 72π
2. Flat circular face. A solid hemisphere has a flat circle on the bottom. Area of a circle = πr².
   π × 6² = 36π
3. Add them. Add the numbers in front of π.
   72π + 36π = 108π
4. 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.
   108π cm²

Check. 3πr² = 3 × π × 36 = 108π. ✓

Watch out. Do not forget the flat circular face of a solid hemisphere: 2πr² + πr² = 3πr².

Key words. The apex is the top point of a pyramid. The vertical height is measured straight down from the apex to the base. The slant height is the height of a sloping triangular face.

Method. Volume of a pyramid = one third × area of the base × vertical height.

1. Write down the formula. A pyramid takes up exactly one third of the prism with the same base and the same height.
   Volume = ⅓ × base area × vertical height
2. Find the area of the base. The base is a rectangle 8 cm by 6 cm.
   8 × 6 = 48 cm²
3. Multiply by the vertical height. Use the vertical height, from the top point straight down to the base (not the slanting edge).
   48 × 9 = 432
4. Divide by 3. Taking one third means dividing by 3.
   432 ÷ 3 = 144
5. Write the unit.
   V = 144 cm³

Check. The prism with the same base and height has volume 432. Three pyramids fill it: 3 × 144 = 432. ✓

Watch out. Do not forget the ⅓. Use the vertical (perpendicular) height, not the sloping edge or slant height.

Method. Split the solid into a cylinder and a hemisphere. Work out the two volumes and add them.

1. Identify the parts. The solid is a cylinder (radius 3 cm, height 7 cm) with a hemisphere of radius 3 cm on top.
2. Volume of the cylinder. Volume of a cylinder = πr²h.
   π × 3² × 7 = π × 9 × 7 = 63π
3. Volume of the hemisphere. Half a sphere: ½ × &frac43;πr³ = ⅔πr³.
   ⅔ × π × 3³ = ⅔ × 27π = 18π
4. Add the parts. The whole solid is made of these parts, so add them. Keep the π terms together.
   63π + 18π = 81π
5. Write the final 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.
   81π cm³

Check. Estimate with π ≈ 3: the total is about 243, close to the answer 254. ✓

Key words. Two solids are similar if one is an enlargement of the other: every length is multiplied by the same scale factor. See Chapter 21 for more on similar shapes.

Method. For similar solids with length scale factor k: areas scale by k² and volumes by k³.

1. Find the length scale factor. Divide a length on the solid we are going to by the matching length on the solid we are going from.
   10 ÷ 6 = 123
2. Find the volume scale factor. Areas are multiplied by the scale factor squared, and volumes (and masses of the same material) by the scale factor cubed. So we raise 123 to the power 3.
   123³ = 123 × 123 × 123 = 41727
3. Multiply the volume. Multiply the known amount by the scale factor.
   108 × 41727 = 500
4. Write the unit.
   500 cm³

Check. Reverse it: 500 ÷ 41727 = 108. ✓

Watch out. Do not multiply the volume by just the length scale factor 123: you must use 123³.

Key words. A frustum is what is left of a cone when the top is cut off by a plane parallel to the base. Similar shapes have the same shape but different sizes: all their lengths are in the same ratio.

Method. Frustum volume = volume of the large cone − volume of the small cone that was cut off.

1. Imagine the missing top of the cone. A frustum is a cone with a smaller cone cut off its top. If we put the small cone back we get a whole large cone.
   Large cone: radius 6, height 4 + x.  Small cone: radius 3, height x.
2. Use similar triangles to find x. The two cones have the same shape, so (radius ÷ height) is the same for both.
   3 ÷ x = 6 ÷ (4 + x)
3. Cross-multiply and solve. Multiply both sides to remove the fractions.
   3(4 + x) = 6x
   12 + 3x = 6x
   12 = 3x
   x = 4
4. Heights of the two cones. The large cone is the frustum height plus x.
   Large cone height = 4 + 4 = 8 cm
   Small cone height = 4 cm
5. Volume of the large cone. V = ⅓πr²h.
   ⅓ × π × 6² × 8 = 96π
6. Volume of the small cone. Same formula with the smaller measurements.
   ⅓ × π × 3² × 4 = 12π
7. Subtract. The frustum is what is left when the small cone is removed.
   96π − 12π = 84π
8. 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 = 84π cm³

Check. Formula check: ⅓πh(R² + Rr + r²) = ⅓ × π × 4 × (36 + 18 + 9) = ⅓ × 252π = 84π. ✓

Watch out. Subtract the volumes (not the heights or radii): large cone − small cone.