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Exam questions · Maths · Geometry and Measures

Volume and Surface Area

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Work out [5 marks]

    The diagram shows a cylinder. Give your answers in terms of \(\pi\). (a) Work out the volume of the cylinder. [2 marks] (b) Work out the total surface area of the cylinder. [3 marks]

    A cylinder with radius 5 cm and height 12 cm.
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    Model answer

    (a) \(\pi \times 5^2 \times 12 = 300\pi\) cm\(^3\). (b) The curved surface is \(2 \times \pi \times 5 \times 12 = 120\pi\) and the two circles are \(2 \times \pi \times 5^2 = 50\pi\). The total is \(170\pi\) cm\(^2\).

    Mark scheme

    • (a) \(\pi \times 5^2 \times 12\) — M1
    • (a) \(300\pi\) — A1
    • (b) \(2\pi \times 5 \times 12\) or \(120\pi\) — M1
    • (b) \(2 \times \pi \times 5^2\) or \(50\pi\) — M1
    • (b) \(170\pi\) — A1
  2. 2 Work out [2 marks]

    A cuboid measures 7 cm by 4 cm by 5 cm. Work out the volume of the cuboid.

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    Model answer

    \(7 \times 4 \times 5 = 140\) cm\(^3\).

    Mark scheme

    • \(7 \times 4 \times 5\) — M1
    • 140 cm\(^3\) — A1
  3. 3 Work out [2 marks]

    A prism has a cross-section of area 18 cm\(^2\) and a volume of 270 cm\(^3\). Work out the length of the prism.

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    Model answer

    Volume \(=\) area \(\times\) length, so the length is \(270 \div 18 = 15\) cm.

    Mark scheme

    • \(270 \div 18\) — M1
    • 15 cm — A1
  4. 4 Work out [3 marks]

    The total surface area of a cube is 150 cm\(^2\). Work out the volume of the cube.

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    Model answer

    A cube has 6 faces, so one face is \(150 \div 6 = 25\) cm\(^2\). The side is \(\sqrt{25} = 5\) cm and the volume is \(5^3 = 125\) cm\(^3\).

    Mark scheme

    • \(150 \div 6 = 25\) — M1
    • Side \(= 5\) — M1
    • 125 cm\(^3\) — A1
  5. 5 Work out [3 marks]

    A solid metal cylinder has radius 2 cm and height 5 cm. The density of the metal is 8 g/cm\(^3\). Work out the mass of the cylinder. Give your answer in terms of \(\pi\).

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    Model answer

    The volume is \(\pi \times 2^2 \times 5 = 20\pi\) cm\(^3\). Mass \(=\) density \(\times\) volume \(= 8 \times 20\pi = 160\pi\) g.

    Mark scheme

    • \(\pi \times 2^2 \times 5\) or \(20\pi\) — M1
    • \(8 \times 20\pi\) — M1
    • \(160\pi\) g — A1
  6. 6 Work out [4 marks]

    A cone has radius 9 cm and vertical height 12 cm. Give your answers in terms of \(\pi\). (a) Work out the volume of the cone. [2 marks] (b) The slant height of the cone is 15 cm. Work out the curved surface area of the cone. [2 marks]

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    Model answer

    (a) \(\dfrac{1}{3} \times \pi \times 9^2 \times 12 = \dfrac{1}{3} \times 972\pi = 324\pi\) cm\(^3\). (b) The curved surface area is \(\pi r l = \pi \times 9 \times 15 = 135\pi\) cm\(^2\).

    Mark scheme

    • (a) \(\dfrac{1}{3} \times \pi \times 9^2 \times 12\) — M1
    • (a) \(324\pi\) — A1
    • (b) \(\pi \times 9 \times 15\) — M1
    • (b) \(135\pi\) — A1

Quick check

  1. 1

    A cuboid measures 8 cm by 5 cm by 3 cm. What is its volume?

    1. A16 cm\(^3\)
    2. B158 cm\(^3\)
    3. C240 cm\(^3\)
    4. D120 cm\(^3\)
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    D: 120 cm\(^3\)

    \(8 \times 5 \times 3 = 120\) cm\(^3\).

  2. 2

    The cross-section of a prism has area 12 cm\(^2\). The prism is 15 cm long. What is its volume?

    1. A27 cm\(^3\)
    2. B90 cm\(^3\)
    3. C180 cm\(^3\)
    4. D12 cm\(^3\)
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    C: 180 cm\(^3\)

    Volume \(=\) area of cross-section \(\times\) length \(= 12 \times 15 = 180\).

  3. 3

    A cylinder has radius 3 cm and height 10 cm. What is its volume, in terms of \(\pi\)?

    1. A\(30\pi\) cm\(^3\)
    2. B\(90\pi\) cm\(^3\)
    3. C\(60\pi\) cm\(^3\)
    4. D\(900\pi\) cm\(^3\)
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    B: \(90\pi\) cm\(^3\)

    \(\pi r^2 h = \pi \times 9 \times 10 = 90\pi\).

  4. 4

    A cylinder has radius 3 cm and height 10 cm. What is its curved surface area, in terms of \(\pi\)?

    1. A\(60\pi\) cm\(^2\)
    2. B\(30\pi\) cm\(^2\)
    3. C\(90\pi\) cm\(^2\)
    4. D\(180\pi\) cm\(^2\)
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    A: \(60\pi\) cm\(^2\)

    \(2\pi r h = 2 \times \pi \times 3 \times 10 = 60\pi\).

  5. 5

    What is the total surface area of a cube with side 4 cm?

    1. A64 cm\(^2\)
    2. B16 cm\(^2\)
    3. C24 cm\(^2\)
    4. D96 cm\(^2\)
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    D: 96 cm\(^2\)

    A cube has 6 faces, each of area \(4 \times 4 = 16\). So \(6 \times 16 = 96\).

  6. 6

    A tank holds 2500 cm\(^3\) of water. How many litres is this?

    1. A25 litres
    2. B0.25 litres
    3. C2.5 litres
    4. D250 litres
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    C: 2.5 litres

    \(1000\text{ cm}^3 = 1\) litre, so \(2500 \div 1000 = 2.5\).

  7. 7

    A cylinder has radius 2 cm and height 7 cm. What is its volume, in terms of \(\pi\)?

    1. A\(14\pi\) cm\(^3\)
    2. B\(28\pi\) cm\(^3\)
    3. C\(56\pi\) cm\(^3\)
    4. D\(98\pi\) cm\(^3\)
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    B: \(28\pi\) cm\(^3\)

    \(\pi \times 2^2 \times 7 = 28\pi\). Using the diameter instead of the radius would give \(98\pi\).

  8. 8

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

    1. A\(36\pi\) cm\(^3\)
    2. B\(12\pi\) cm\(^3\)
    3. C\(27\pi\) cm\(^3\)
    4. D\(108\pi\) cm\(^3\)
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    A: \(36\pi\) cm\(^3\)

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

  9. 9

    A cone has radius 3 cm and vertical height 4 cm. What is its volume, in terms of \(\pi\)?

    1. A\(36\pi\) cm\(^3\)
    2. B\(15\pi\) cm\(^3\)
    3. C\(9\pi\) cm\(^3\)
    4. D\(12\pi\) cm\(^3\)
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    D: \(12\pi\) cm\(^3\)

    \(\dfrac{1}{3}\pi r^2 h = \dfrac{1}{3} \times \pi \times 9 \times 4 = 12\pi\). Using the slant height 5 gives \(15\pi\), which is wrong.