The frustum of a cone can be thought of as a cone with the top missing.
The volume of a frustum of a cone can be thought of as the difference between two cones' volumes.
This formula can be manipulated to rely on information from the frustum only.

Common misconception

Pupils may think that if you cut the height of the cone in a half, the resulting frustrum will have half the volume of the original cone.

The frustrum is actually seven eighths of the volume of the original cone; the small cone that was removed to create the frustrum is one eighth the volume of the original cone if the frustrum and the small cone have the same height.

Keywords

Frustum - A frustum is the 3D shape made from a cone by making a cut parallel to its circular base and removing the resultant smaller cone.
Volume - Volume is the amount of space occupied by a closed 3D shape.

Links can be made to conversions between metric units of length, units of area or units of volume. This topic also provides opportunities to practice using Pythagoras' theorem or trigonometry in context.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.

A solid can be decomposed to make the volume easier to calculate.

Correct Answer: composite, compound

Q2.

The diagram shows a composite solid constructed from two congruent cuboids. All lengths given are in centimetres. Which of these calculations give the total volume of the solid?

Correct answer: (4 × 16 + 8 × 4) × 6

4 × 6 × 16 + 12 × 6 × 4

Correct answer: 4 × 16 × 6 + 8 × 6 × 4

16 × 12 × 6

Correct answer: 2 × (12 × 4 × 6)

Q3.

This composite solid is constructed from two cuboids. All lengths given are in metres. The total volume of the solid is m³.

Correct Answer: 880

Q4.

This composite solid is constructed by placing a hemisphere with diameter 10 cm on top of a cuboid. Find the volume of the solid. Give your answer to the nearest cubic centimetre.

652 cm³

Correct answer: 862 cm³

1124 cm³

2694 cm³

Q5.

This composite solid is constructed with a cylinder and a hemisphere. Each have a diameter of 12 cm. The volume of the solid is cm³ (correct to 4 significant figures).

Correct Answer: 2149

Q6.

This composite solid is constructed with a cone and a hemisphere. The volume of the solid, in terms of 𝜋, is 𝜋 cm³.

Correct Answer: 264

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6 Questions

Q1.

A is the 3D shape made from a cone by making a cut parallel to its circular base and removing the resultant smaller cone.

Correct Answer: frustum

Q2.

The left-hand cone is split into a smaller cone and a frustum. Match each length labelled a - c to its value.

Correct Answer:a,15 cm

15 cm

Correct Answer:b,5 cm

5 cm

Correct Answer:c,20 cm

20 cm

Q3.

The diagram shows a side elevation of a frustum. Find the volume of the large cone that the frustum is cut from. Give your answer correct to 3 significant figures.

3421 cm³ to 3 s.f.

9120 cm³ to 3 s.f.

Correct answer: 12 500 cm³ to 3 s.f.

50 200 cm³ to 3 s.f.

Q4.

The diagram shows a side elevation of a frustum. Find the volume of the frustum. Give your answer correct to 3 significant figures.

4294 cm³

Correct answer: 7720 cm³

8790 cm³

12 500 cm³

Q5.

A cone is cut into a frustum of height 20 cm and a small cone. The radius of the base of the frustum is 15 cm and the radius of its top is 9 cm. The height of the small cone is cm.

Correct Answer: 30

Q6.

A cone is cut into a frustum of height 20 cm and a small cone. The radius of the base of the frustum is 15 cm and the radius of its top is 9 cm. The volume of the frustum is 𝜋 cm

Correct Answer: 2940, 2 940, 2,940