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Maths · Area and volume

Pyramids and cones

Work out the volume of pyramids and cones and the curved surface area of a cone, and at Higher tier the volume of a frustum.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the area of a square with side 6?

    Show answerHide answer

    36

  2. 2

    State Pythagoras' theorem.

    Show answerHide answer

    \(a^2 + b^2 = c^2\)

  3. 3

    What is \(\dfrac{1}{3}\) of 36 multiplied by 9?

    Show answerHide answer

    108

  4. 4

    What is the area of a circle with radius 4, in terms of \(\pi\)?

    Show answerHide answer

    \(16\pi\)

  5. 5

    What is the volume of a prism?

    Show answerHide answer

    Cross-sectional area times length

Learning Objectives

  1. 1Find the volume of a pyramid using \(\dfrac{1}{3} \times \text{base area} \times h\).
  2. 2Find the volume and curved surface area of a cone.
  3. 3Use Pythagoras to find a slant height.
  4. 4Find the volume of a frustum (Higher).

THE KEY FACT

A pyramid or cone has one third of the volume of the prism or cylinder with the same base and height.

\(V = \dfrac{1}{3} \times \text{base area} \times \text{perpendicular height}\)

A Pyramid

A square-based pyramid has base side 6 cm and perpendicular height 9 cm. Find its volume.

Show the solutionHide the solution
  1. 1 Area of the base \(6 \times 6 = 36\)
  2. 2 Volume \(= \dfrac{1}{3} \times \text{base area} \times h\) \(\dfrac{1}{3} \times 36 \times 9\)
  3. 3 Work out \(108\)

Answer108 cm³

A Cone

A cone has base radius 4 cm and perpendicular height 9 cm. Find its volume in terms of \(\pi\) and to 1 decimal place.

Show the solutionHide the solution
  1. 1 Volume \(= \dfrac{1}{3}\pi r^2 h\) \(\dfrac{1}{3} \times \pi \times 16 \times 9\)
  2. 2 Simplify \(48\pi\)
  3. 3 As a decimal \(150.8\)

Answer\(48\pi = 150.8\) cm³

Slant Height and Curved Surface

A cone has base radius 5 cm and perpendicular height 12 cm. Find (a) the slant height, and (b) the curved surface area in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 (a) The radius, height and slant height form a right-angled triangle \(l^2 = 5^2 + 12^2 = 169\)
  2. 2 Slant height \(l = 13\)
  3. 3 (b) Curved surface area \(= \pi r l\) \(\pi \times 5 \times 13 = 65\pi\)

Answer(a) 13 cm (b) \(65\pi\) cm² (204.2 cm²)

Formulae for Pyramids and Cones

The volume formula for a cone is on the formulae sheet; the curved surface area of a cone is too.

  • Pyramid

    Volume: \(\dfrac{1}{3} \times \text{base area} \times h\). Surface: Add the areas of all the faces

  • Cone

    Volume: \(\dfrac{1}{3}\pi r^2 h\). Surface: Curved surface \(\pi r l\); base \(\pi r^2\)

  • Slant height

    Volume: \(l = \sqrt{r^2 + h^2}\). Surface: Pythagoras in the cone

Volume of a Frustum

A cone of radius 6 cm and height 12 cm has a small cone of radius 3 cm and height 6 cm cut off the top. Find the volume of the frustum that is left, in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Volume of the whole cone \(\dfrac{1}{3} \times \pi \times 6^2 \times 12 = 144\pi\)
  2. 2 Volume of the small cone \(\dfrac{1}{3} \times \pi \times 3^2 \times 6 = 18\pi\)
  3. 3 Subtract \(144\pi - 18\pi\)

Answer\(126\pi\) cm³ (395.8 cm³)

Ice Cream Cone

A cone has radius 3 cm and slant height 10 cm. Work out the perpendicular height, the volume of the cone, and the volume of a hemisphere of ice cream of radius 3 cm on top. Will the ice cream fit inside the cone if it melts?

1. Use Pythagoras for the height.

2. Use each volume formula.

3. Compare.

A good answer shows: Height \(= \sqrt{100 - 9} = 9.54\) cm. Cone volume \(= \dfrac{1}{3}\pi \times 9 \times 9.54 = 89.9\) cm³. Hemisphere \(= \dfrac{2}{3}\pi \times 27 = 56.5\) cm³. The ice cream volume is less than the cone volume, so it would fit.

Can I...?

  1. 1Find the volume of a pyramid.
  2. 2Find the volume of a cone.
  3. 3Use Pythagoras to find a slant height.
  4. 4Find the curved surface area of a cone.
  5. 5Find the total surface area of a cone.
  6. 6Work backwards to find a height.
  7. 7Find the volume of a frustum (Higher).
  8. 8Give answers in terms of \(\pi\).

Summary & Exam Focus

  • Pyramid and cone volume: one third of base area times perpendicular height.
  • Cone curved surface \(\pi r l\), with \(l = \sqrt{r^2 + h^2}\).
  • Do not confuse perpendicular height with slant height.
  • Frustum: whole cone minus small cone (Higher).

Exam focus

A cone has radius 5 cm and vertical height 12 cm. Work out the curved surface area of the cone. Give your answer in terms of \(\pi\). (3 marks) (3 marks)

Find the slant height with Pythagoras before you use \(\pi r l\). The vertical height goes in the volume formula; the slant height goes in the surface area formula.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Pyramid
A solid with a polygon base and triangular faces meeting at a point.
Apex
The point at the top of a pyramid or cone.
Cone
A solid with a circular base and a curved surface up to an apex.
Perpendicular height
The height measured at right angles to the base.
Slant height
The distance from the apex down the side of a cone to the base edge.
Frustum
What is left when the top of a cone or pyramid is cut off parallel to the base.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 2 marks Easier

A square-based pyramid has a base of side 6 cm and a perpendicular height of 9 cm. Work out the volume of the pyramid.

Mark scheme — 2 marks available

  • \(\dfrac{1}{3} \times 6 \times 6 \times 9\) — M1
  • 108 — A1

Model answer

\(\dfrac{1}{3} \times 36 \times 9 = 108\) cm³.

2. Exam question Calculator 2 marks Easier

A cone has a base radius of 4 cm and a perpendicular height of 9 cm. Work out the volume of the cone. Give your answer correct to 3 significant figures.

Mark scheme — 2 marks available

  • \(\dfrac{1}{3}\pi \times 4^2 \times 9\) — M1
  • 151 — A1

Model answer

\(\dfrac{1}{3}\pi \times 4^2 \times 9 = 48\pi = 151\) cm³.

3. Exam question Non-calculator 3 marks Easier

The diagram shows a cone with base radius 5 cm and vertical height 12 cm. Work out the curved surface area of the cone. Give your answer in terms of \(\pi\).

A cone with base radius 5 cm and vertical height 12 cm; the slant height is not given.

Mark scheme — 3 marks available

  • \(5^2 + 12^2\) — M1
  • Slant height 13 — A1
  • \(65\pi\) — A1

Model answer

The slant height is \(\sqrt{5^2 + 12^2} = 13\) cm. The curved surface area is \(\pi \times 5 \times 13 = 65\pi\) cm².

4. Exam question Calculator 2 marks Easier

Work out the volume of the cone in the last question. Give your answer correct to 3 significant figures.

Mark scheme — 2 marks available

  • \(\dfrac{1}{3}\pi \times 5^2 \times 12\) — M1
  • 314 — A1

Model answer

\(\dfrac{1}{3}\pi \times 5^2 \times 12 = 100\pi = 314\) cm³.

5. Exam question Calculator 4 marks Easier

A cone has base radius 6 cm and height 12 cm. A smaller cone of base radius 3 cm and height 6 cm is cut off the top, leaving a frustum. Work out the volume of the frustum. Give your answer correct to 3 significant figures.

Mark scheme — 4 marks available

  • Volume of the large cone — M1
  • Volume of the small cone — M1
  • Subtracting — M1
  • 396 — A1

Model answer

Large cone: \(\dfrac{1}{3}\pi \times 6^2 \times 12 = 144\pi\). Small cone: \(\dfrac{1}{3}\pi \times 3^2 \times 6 = 18\pi\). Frustum: \(126\pi = 396\) cm³.

6. Exam question Non-calculator 2 marks Easier

A pyramid has a rectangular base measuring 8 cm by 6 cm and a perpendicular height of 5 cm. Work out the volume of the pyramid.

Mark scheme — 2 marks available

  • \(\dfrac{1}{3} \times 8 \times 6 \times 5\) — M1
  • 80 — A1

Model answer

\(\dfrac{1}{3} \times 8 \times 6 \times 5 = 80\) cm³.

7. Multiple choice 1 mark Easier

What fraction of a prism's volume is a pyramid with the same base and height?

  1. A One half
  2. B One quarter
  3. C One third Correct
  4. D Two thirds

Why: A pyramid has one third of the volume.

8. Multiple choice 1 mark Core

A cone has radius 3 cm and height 4 cm. What is the slant height?

  1. A 4 cm
  2. B 5 cm Correct
  3. C 7 cm
  4. D 25 cm

Why: \(\sqrt{9 + 16} = 5\).

9. Multiple choice 1 mark Core

What is the volume of a cone with radius 3 cm and height 4 cm, in terms of \(\pi\)?

  1. A \(12\pi\) Correct
  2. B \(36\pi\)
  3. C \(4\pi\)
  4. D \(15\pi\)

Why: \(\dfrac{1}{3}\pi \times 9 \times 4 = 12\pi\).

10. Multiple choice 1 mark Core

Which formula gives the curved surface area of a cone?

  1. A \(\pi r^2\)
  2. B \(\pi r h\)
  3. C \(2\pi r\)
  4. D \(\pi r l\) Correct

Why: Curved surface area is \(\pi r l\), where \(l\) is the slant height.

11. Multiple choice 1 mark Core

A square pyramid has base area 25 cm² and height 6 cm. What is its volume?

  1. A 150 cm³
  2. B 50 cm³ Correct
  3. C 25 cm³
  4. D 75 cm³

Why: \(\dfrac{1}{3} \times 25 \times 6 = 50\).

12. Multiple choice 1 mark Stretch

A frustum is made by cutting a small cone off a big cone. Its volume is...

  1. A The small cone plus the big cone
  2. B The big cone divided by the small cone
  3. C The big cone minus the small cone Correct
  4. D Half the big cone

Why: Volume of the big cone minus the volume of the small cone.