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Exam questions · Maths · Geometry and Measures

Volume and Surface Area

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

    The diagram shows a prism. The cross-section of the prism is a right-angled triangle. (a) Work out the volume of the prism. (2 marks) (b) Work out the total surface area of the prism. (3 marks)

    A triangular prism whose cross-section is a 3, 4, 5 right-angled triangle and whose length is 10 cm.
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    Model answer

    (a) The area of the triangle is \(\dfrac{1}{2} \times 4 \times 3 = 6\) cm\(^2\), so the volume is \(6 \times 10 = 60\) cm\(^3\). (b) The two triangles have area \(2 \times 6 = 12\). The three rectangles have areas \(4 \times 10 = 40\), \(3 \times 10 = 30\) and \(5 \times 10 = 50\), which total 120. The total surface area is \(12 + 120 = 132\) cm\(^2\).

    Mark scheme

    • (a) \(\dfrac{1}{2} \times 4 \times 3 = 6\) — M1
    • (a) 60 cm\(^3\) — A1
    • (b) \(40\), \(30\) and \(50\) found, or \(12 \times 10\) — M1
    • (b) \(12\) for the triangles added to their rectangles — M1
    • (b) 132 cm\(^2\) — A1
  2. 2 Work out [2 marks]

    A cuboid has a volume of 240 cm\(^3\). Its length is 10 cm and its width is 6 cm. Work out its height.

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    Model answer

    \(10 \times 6 \times h = 240\), so \(60h = 240\) and \(h = 4\) cm.

    Mark scheme

    • \(240 \div (10 \times 6)\) or \(60h = 240\) — M1
    • 4 cm — A1
  3. 3 Work out [4 marks]

    A cylinder has radius 5 cm and height 8 cm. Give your answers in terms of \(\pi\). (a) Work out the volume of the cylinder. (2 marks) (b) Work out the curved surface area of the cylinder. (2 marks)

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    Model answer

    (a) \(\pi \times 5^2 \times 8 = 200\pi\) cm\(^3\). (b) \(2 \times \pi \times 5 \times 8 = 80\pi\) cm\(^2\).

    Mark scheme

    • (a) \(\pi \times 5^2 \times 8\) — M1
    • (a) \(200\pi\) — A1
    • (b) \(2 \times \pi \times 5 \times 8\) — M1
    • (b) \(80\pi\) — A1
  4. 4 Work out [3 marks]

    A tank is in the shape of a cuboid measuring 80 cm by 50 cm by 40 cm. The tank is full of water. Work out the volume of water in the tank in litres.

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    Model answer

    \(80 \times 50 \times 40 = 160\,000\) cm\(^3\). Since \(1000\text{ cm}^3 = 1\) litre, the volume is \(160\) litres.

    Mark scheme

    • \(80 \times 50 \times 40\) — M1
    • \(160\,000\) cm\(^3\) — A1
    • 160 litres — A1
  5. 5 Show that [3 marks]

    A cylinder has radius \(x\) cm and height \(2x\) cm. Show that the volume of the cylinder is \(2\pi x^3\) cm\(^3\).

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    Model answer

    Volume \(= \pi r^2 h = \pi \times x^2 \times 2x = 2\pi x^3\), as required.

    Mark scheme

    • \(\pi \times x^2 \times h\) with \(h = 2x\) substituted — M1
    • \(\pi x^2 \times 2x\) — M1
    • \(2\pi x^3\) with a conclusion — C1
  6. 6 Work out [4 marks]

    Give your answers in terms of \(\pi\). (a) A sphere has radius 6 cm. Work out the volume of the sphere. (2 marks) (b) Work out the curved surface area of a hemisphere of radius 6 cm. (2 marks)

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    Model answer

    (a) \(\dfrac{4}{3} \times \pi \times 6^3 = \dfrac{4}{3} \times 216\pi = 288\pi\) cm\(^3\). (b) The surface area of a sphere is \(4\pi r^2 = 144\pi\), so half of it is \(72\pi\) cm\(^2\).

    Mark scheme

    • (a) \(\dfrac{4}{3} \times \pi \times 216\) — M1
    • (a) \(288\pi\) — A1
    • (b) \(\dfrac{1}{2} \times 4\pi \times 36\) or \(2\pi r^2\) — M1
    • (b) \(72\pi\) — 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.