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Maths · Area and volume

Cylinders and spheres

Work out the volume and surface area of cylinders, spheres and hemispheres, and solve problems that combine solids.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

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

    Show answerHide answer

    \(16\pi\)

  2. 2

    What is the circumference of a circle with diameter 10, in terms of \(\pi\)?

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    \(10\pi\)

  3. 3

    Work out \(5^3\).

    Show answerHide answer

    125

  4. 4

    What is the volume of a prism?

    Show answerHide answer

    Cross-sectional area times length

  5. 5

    Round 3.979 to 2 decimal places.

    Show answerHide answer

    3.98

Learning Objectives

  1. 1Find the volume and surface area of a cylinder.
  2. 2Find the volume and surface area of a sphere.
  3. 3Find the volume and surface area of a hemisphere.
  4. 4Solve problems that combine cylinders and hemispheres.

Formulae

The sphere formulae are given on the Edexcel formulae sheet, but you must know the cylinder ones.

  • Cylinder

    Volume: \(\pi r^2 h\). Surface area: \(2\pi r^2 + 2\pi r h\)

  • Sphere

    Volume: \(\dfrac{4}{3}\pi r^3\). Surface area: \(4\pi r^2\)

  • Hemisphere

    Volume: \(\dfrac{2}{3}\pi r^3\). Surface area: \(3\pi r^2\)

A Cylinder

A cylinder has radius 5 cm and height 12 cm. Find its volume and total surface area, in terms of \(\pi\) and to 1 decimal place.

Show the solutionHide the solution
  1. 1 Volume \(= \pi r^2 h\) \(\pi \times 25 \times 12 = 300\pi\)
  2. 2 As a decimal \(942.5\) cm³
  3. 3 Curved surface \(= 2\pi r h\) \(2\pi \times 5 \times 12 = 120\pi\)
  4. 4 Two circular ends \(= 2\pi r^2\) \(2\pi \times 25 = 50\pi\)
  5. 5 Total surface area \(170\pi = 534.1\) cm²

AnswerVolume \(300\pi\) cm³ (942.5) and surface area \(170\pi\) cm² (534.1).

A Sphere

A sphere has radius 6 cm. Find its volume and surface area, in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Volume \(= \dfrac{4}{3}\pi r^3\) \(\dfrac{4}{3} \times \pi \times 216 = 288\pi\)
  2. 2 Surface area \(= 4\pi r^2\) \(4 \times \pi \times 36 = 144\pi\)

AnswerVolume \(288\pi\) cm³ and surface area \(144\pi\) cm².

A Hemisphere

A solid hemisphere has radius 3 cm. Find its volume in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 A hemisphere is half a sphere \(\dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3\)
  2. 2 Substitute \(r = 3\) \(\dfrac{2}{3} \times \pi \times 27\)
  3. 3 Work out \(18\pi\)

Answer\(18\pi\) cm³

Finding the Height

A cylinder has volume 200 cm³ and radius 4 cm. Find its height to 2 decimal places.

Show the solutionHide the solution
  1. 1 \(\pi r^2 h = 200\) \(\pi \times 16 \times h = 200\)
  2. 2 Divide by \(16\pi\) \(h = \dfrac{200}{16\pi}\)
  3. 3 Calculate \(3.979...\)

Answer3.98 cm

Combined Solids

Add the volumes; for surface area, do not count the joined faces.

  • A capsule

    A cylinder with a hemisphere at each end: total volume \(= \pi r^2 h + \dfrac{4}{3}\pi r^3\).

  • A hemisphere on top

    Surface area is the cylinder's curved surface and one end, plus the hemisphere's curved surface \(2\pi r^2\).

  • Watch the radius

    The radius of the hemisphere is the radius of the cylinder.

Tennis Ball Tube

Three tennis balls of radius 3.3 cm fit exactly in a cylindrical tube. Find the volume of the tube, the volume of the three balls, and the percentage of the tube filled with balls.

1. Find the height of the tube.

2. Work out both volumes.

3. Divide and convert to a percentage.

A good answer shows: The tube has radius 3.3 and height 19.8: \(\pi \times 3.3^2 \times 19.8 = 677.4\) cm³. The balls: \(3 \times \dfrac{4}{3}\pi \times 3.3^3 = 4\pi \times 35.937 = 451.6\) cm³. Percentage \(= 451.6 \div 677.4 = 66.7\%\), exactly two thirds.

Can I...?

  1. 1Find the volume of a cylinder.
  2. 2Find the surface area of a cylinder.
  3. 3Find the volume of a sphere.
  4. 4Find the surface area of a sphere.
  5. 5Find the volume of a hemisphere.
  6. 6Work backwards to find a radius or height.
  7. 7Solve problems with combined solids.
  8. 8Give answers in terms of \(\pi\).

Summary & Exam Focus

  • Cylinder volume \(\pi r^2 h\); curved surface \(2\pi r h\).
  • Sphere volume \(\dfrac{4}{3}\pi r^3\); surface \(4\pi r^2\).
  • Hemisphere volume \(\dfrac{2}{3}\pi r^3\).
  • For combined solids, add volumes and take care with joined faces.

Exam focus

A cylinder has radius 5 cm and height 12 cm. Work out the volume of the cylinder. Give your answer correct to 3 significant figures. (2 marks) (2 marks)

Cylinder volume is the area of the circle times the height: \(\pi r^2 h\). Check that you have the radius, not the diameter.

Key terms

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

Cylinder
A prism with a circle as its cross-section.
Sphere
A perfectly round 3D shape, like a ball.
Hemisphere
Half a sphere.
Curved surface area
The area of the curved part only, not the flat ends.
Volume
The amount of space inside a solid.
In terms of \(\pi\)
Leaving \(\pi\) in the answer.

Practice questions

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

  1. Question 1 Calculator 2 marks

    The diagram shows a cylinder with radius 5 cm and height 12 cm. Work out the volume of the cylinder. Give your answer correct to 3 significant figures.

    A cylinder with radius 5 cm and height 12 cm.
    Show answerHide answer

    Model answer

    \(\pi \times 5^2 \times 12 = 300\pi = 942\) cm³.

    Mark scheme

    • \(\pi \times 5^2 \times 12\) — M1
    • 942 — A1
  2. Question 2 Calculator 3 marks

    Work out the total surface area of the cylinder in the last question. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    \(2\pi \times 5^2 + 2\pi \times 5 \times 12 = 50\pi + 120\pi = 170\pi = 534\) cm².

    Mark scheme

    • \(2\pi r^2\) — M1
    • \(2\pi r h\) — M1
    • 534 — A1
  3. Question 3 Calculator 2 marks

    A sphere has a radius of 6 cm. Work out the volume of the sphere. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    \(\dfrac{4}{3}\pi \times 6^3 = 288\pi = 905\) cm³.

    Mark scheme

    • \(\dfrac{4}{3}\pi \times 6^3\) — M1
    • 905 — A1
  4. Question 4 Calculator 2 marks

    Work out the surface area of the sphere in the last question. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    \(4\pi \times 6^2 = 144\pi = 452\) cm².

    Mark scheme

    • \(4\pi \times 6^2\) — M1
    • 452 — A1
  5. Question 5 Calculator 3 marks

    A cylinder has a volume of 200 cm³ and a radius of 4 cm. Work out the height of the cylinder. Give your answer correct to 2 decimal places.

    Show answerHide answer

    Model answer

    \(\pi \times 4^2 \times h = 200\), so \(h = \dfrac{200}{16\pi} = 3.98\) cm.

    Mark scheme

    • \(\pi \times 4^2 \times h = 200\) — M1
    • \(\dfrac{200}{16\pi}\) — M1
    • 3.98 — A1
  6. Question 6 Non-calculator 2 marks

    A solid hemisphere has a radius of 3 cm. Work out the volume of the hemisphere. Give your answer in terms of \(\pi\).

    Show answerHide answer

    Model answer

    \(\dfrac{2}{3}\pi \times 3^3 = 18\pi\) cm³.

    Mark scheme

    • \(\dfrac{2}{3}\pi \times 27\) or half of \(\dfrac{4}{3}\pi \times 27\) — M1
    • \(18\pi\) — A1

Quick check

  1. Which formula gives the volume of a cylinder?

    1. A\(2\pi r h\)
    2. B\(\pi r^2 h\)
    3. C\(\dfrac{4}{3}\pi r^3\)
    4. D\(\pi r h\)
    Show answerHide answer

    B: \(\pi r^2 h\)

    The area of the circle times the height: \(\pi r^2 h\).

  2. A cylinder has radius 2 cm and height 5 cm. What is its volume in terms of \(\pi\)?

    1. A\(10\pi\)
    2. B\(40\pi\)
    3. C\(20\pi\)
    4. D\(50\pi\)
    Show answerHide answer

    C: \(20\pi\)

    \(\pi \times 4 \times 5 = 20\pi\).

  3. A sphere has radius 3 cm. What is its volume in terms of \(\pi\)?

    1. A\(36\pi\)
    2. B\(12\pi\)
    3. C\(108\pi\)
    4. D\(9\pi\)
    Show answerHide answer

    A: \(36\pi\)

    \(\dfrac{4}{3}\pi \times 27 = 36\pi\).

  4. The curved surface area of a cylinder is...

    1. A\(\pi r^2 h\)
    2. B\(2\pi r^2\)
    3. C\(\pi r h\)
    4. D\(2\pi r h\)
    Show answerHide answer

    D: \(2\pi r h\)

    Unroll it to a rectangle: circumference \(2\pi r\) times height \(h\).

  5. What is the volume of a hemisphere compared with the sphere of the same radius?

    1. ATwice as big
    2. BHalf as big
    3. CA third as big
    4. DThe same
    Show answerHide answer

    B: Half as big

    A hemisphere is half a sphere.

  6. A sphere's radius doubles. What happens to its volume?

    1. AIt doubles
    2. BIt multiplies by 4
    3. CIt multiplies by 8
    4. DIt multiplies by 6
    Show answerHide answer

    C: It multiplies by 8

    Volume is proportional to \(r^3\), so it multiplies by \(2^3 = 8\).

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