GCSE Maths: Volume of prisms (Higher)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Volume of prisms
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.
Method. Write the area of the trapezium using x, form an equation for the volume, and solve it.
- Area of the cross-section. Area of a trapezium = ½ × (sum of parallel sides) × distance between them.Area = ½ × (x + x + 4) × 5 = ½ × (2x + 4) × 5 = (x + 2) × 5 = 5x + 10
- Write the volume equation. Volume = area × length.(5x + 10) × 10 = 350
- Divide both sides by 10. This undoes the multiplication by 10.5x + 10 = 35
- Subtract 10, then divide by 5. Undo the + 10 first, then the × 5.5x = 25, so x = 5
Check. Parallel sides 5 and 9: area = ½ × 14 × 5 = 35, and 35 × 10 = 350. ✓
Watch out. Write the area as an expression in x first. Do not substitute guesses.
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. Find the area of the triangle in terms of x, then multiply by the length.
- Area of the triangle. ½ × base × height.½ × x × 2x = x × x = x2
- Multiply by the length. Volume = x2 × 5 = 5x2 ✓
Check. Try x = 2: base 2, height 4, area 4, volume 20; and 5 × 22 = 20. ✓
Method. Put the volume equal to 500 and solve.
- Write the equation. 5x2 = 500
- Divide by 5. x2 = 100
- Square root. Take the positive value because x is a length.x = 10
Check. 5 × 102 = 500. ✓
Method. Both prisms have the same length, so the ratio of volumes equals the ratio of the cross-section areas.
- Area of the triangle. ½ × 6 × 4.12 cm2
- Write the ratio of areas. The first area is 24 and the second is 12.24 : 12
- Simplify. Divide both parts by 12.24 : 12 = 2 : 1
Check. If both prisms are 10 cm long: volumes 240 and 120, ratio 2 : 1. ✓