OpenRevise GCSE

Viewing as

Student Teacher

Teaching this? The teacher view adds 2 files, the mark schemes and the model answers.

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
  • All boards
Download the full pack · 3 files

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.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    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.

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(\dfrac{1}{3} \times 6 \times 6 \times 9\) — M1
    • 108 — A1
  2. Question 2 Calculator 2 marks

    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.

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(\dfrac{1}{3}\pi \times 4^2 \times 9\) — M1
    • 151 — A1
  3. Question 3 Non-calculator 3 marks

    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.
    Show answerHide answer

    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.

    Mark scheme

    • \(5^2 + 12^2\) — M1
    • Slant height 13 — A1
    • \(65\pi\) — A1
  4. Question 4 Calculator 2 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(\dfrac{1}{3}\pi \times 5^2 \times 12\) — M1
    • 314 — A1
  5. Question 5 Calculator 4 marks

    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.

    Show answerHide answer

    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.

    Mark scheme

    • Volume of the large cone — M1
    • Volume of the small cone — M1
    • Subtracting — M1
    • 396 — A1
  6. Question 6 Non-calculator 2 marks

    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.

    Show answerHide answer

    Model answer

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

    Mark scheme

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

Quick check

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

    1. AOne half
    2. BOne quarter
    3. COne third
    4. DTwo thirds
    Show answerHide answer

    C: One third

    A pyramid has one third of the volume.

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

    1. A4 cm
    2. B5 cm
    3. C7 cm
    4. D25 cm
    Show answerHide answer

    B: 5 cm

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

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

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

    A: \(12\pi\)

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

  4. 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\(\pi r l\)
    Show answerHide answer

    D: \(\pi r l\)

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

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

    1. A150 cm³
    2. B50 cm³
    3. C25 cm³
    4. D75 cm³
    Show answerHide answer

    B: 50 cm³

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

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

    1. AThe small cone plus the big cone
    2. BThe big cone divided by the small cone
    3. CThe big cone minus the small cone
    4. DHalf the big cone
    Show answerHide answer

    C: The big cone minus the small cone

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

Downloads

Free to keep, print and annotate.

Download all 3 as a zip
Something here looks wrong?

Tell us what and we will go and look. It goes to whoever writes these pages, nobody else, and we do not ask who you are — so there is nothing to sign and nothing comes back to you.