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Exam questions · Maths · Geometry and Measures

Area, Perimeter and Circles

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

    The diagram shows a U-shaped metal plate. (a) Work out the area of the plate. (3 marks) (b) Work out the perimeter of the plate. (2 marks)

    A U-shaped compound shape, 12 cm wide and 9 cm tall, with a rectangular gap 4 cm wide and 5 cm deep cut from the top.
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    Model answer

    (a) The whole rectangle is \(12 \times 9 = 108\) cm\(^2\) and the gap is \(4 \times 5 = 20\) cm\(^2\), so the area is \(108 - 20 = 88\) cm\(^2\). (b) The edges are \(12 + 9 + 4 + 5 + 4 + 5 + 4 + 9 = 52\) cm.

    Mark scheme

    • (a) \(12 \times 9\) or \(4 \times 5\) — M1
    • (a) \(108 - 20\) — M1
    • (a) 88 cm\(^2\) — A1
    • (b) All the outside edges added, with at most one error — M1
    • (b) 52 cm — A1
  2. 2 Work out [2 marks]

    A trapezium has parallel sides of length 8 cm and 14 cm. The distance between the parallel sides is 5 cm. Work out the area of the trapezium.

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    Model answer

    Area \(= \dfrac{1}{2}(8 + 14) \times 5 = 11 \times 5 = 55\) cm\(^2\).

    Mark scheme

    • \(\dfrac{1}{2}(8 + 14) \times 5\) or \(11 \times 5\) — M1
    • 55 cm\(^2\) — A1
  3. 3 Work out [2 marks]

    A triangle has a base of 12 cm and an area of 54 cm\(^2\). Work out the perpendicular height of the triangle.

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    Model answer

    \(\dfrac{1}{2} \times 12 \times h = 54\), so \(6h = 54\) and \(h = 9\) cm.

    Mark scheme

    • \(\dfrac{1}{2} \times 12 \times h = 54\) or \(54 \times 2 \div 12\) — M1
    • 9 cm — A1
  4. 4 Work out [4 marks]

    Give your answers to this question in terms of \(\pi\). (a) A circle has radius 6 cm. Work out the area of the circle. (2 marks) (b) A circle has diameter 10 cm. Work out the circumference of the circle. (2 marks)

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    Model answer

    (a) \(\pi \times 6^2 = 36\pi\) cm\(^2\). (b) \(\pi \times 10 = 10\pi\) cm.

    Mark scheme

    • (a) \(\pi \times 6^2\) — M1
    • (a) \(36\pi\) — A1
    • (b) \(\pi \times 10\) or \(2 \times \pi \times 5\) — M1
    • (b) \(10\pi\) — A1
  5. 5 Work out [3 marks]

    The area of a circle is \(49\pi\) cm\(^2\). Work out the circumference of the circle. Give your answer in terms of \(\pi\).

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    Model answer

    \(\pi r^2 = 49\pi\), so \(r^2 = 49\) and \(r = 7\) cm. The circumference is \(2 \times \pi \times 7 = 14\pi\) cm.

    Mark scheme

    • \(r^2 = 49\) or \(r = 7\) — M1
    • \(2 \times \pi \times 7\) — M1
    • \(14\pi\) cm — A1
  6. 6 Work out [4 marks]

    A sector of a circle has radius 12 cm and angle \(150^\circ\). Give your answers in terms of \(\pi\). (a) Work out the length of the arc of the sector. (2 marks) (b) Work out the area of the sector. (2 marks)

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    Model answer

    \(\dfrac{150}{360} = \dfrac{5}{12}\). (a) The circumference is \(2 \times \pi \times 12 = 24\pi\), so the arc is \(\dfrac{5}{12} \times 24\pi = 10\pi\) cm. (b) The area of the circle is \(\pi \times 144 = 144\pi\), so the sector is \(\dfrac{5}{12} \times 144\pi = 60\pi\) cm\(^2\).

    Mark scheme

    • (a) \(\dfrac{150}{360} \times 24\pi\) — M1
    • (a) \(10\pi\) — A1
    • (b) \(\dfrac{150}{360} \times 144\pi\) — M1
    • (b) \(60\pi\) — A1

Quick check

  1. 1

    A triangle has base 10 cm and perpendicular height 6 cm. What is its area?

    1. A60 cm\(^2\)
    2. B16 cm\(^2\)
    3. C30 cm\(^2\)
    4. D32 cm\(^2\)
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    C: 30 cm\(^2\)

    \(\dfrac{1}{2} \times 10 \times 6 = 30\) cm\(^2\).

  2. 2

    A trapezium has parallel sides of 7 cm and 13 cm, and a height of 6 cm. What is its area?

    1. A120 cm\(^2\)
    2. B60 cm\(^2\)
    3. C26 cm\(^2\)
    4. D78 cm\(^2\)
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    B: 60 cm\(^2\)

    \(\dfrac{1}{2}(7 + 13) \times 6 = 60\). Forgetting the half gives 120.

  3. 3

    A parallelogram has base 9 cm, slanted side 5 cm and perpendicular height 4 cm. What is its area?

    1. A36 cm\(^2\)
    2. B45 cm\(^2\)
    3. C18 cm\(^2\)
    4. D20 cm\(^2\)
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    A: 36 cm\(^2\)

    \(\text{base} \times \text{perpendicular height} = 9 \times 4 = 36\). The slanted side is not used.

  4. 4

    A circle has radius 5 cm. What is its area, in terms of \(\pi\)?

    1. A\(10\pi\) cm\(^2\)
    2. B\(5\pi\) cm\(^2\)
    3. C\(50\pi\) cm\(^2\)
    4. D\(25\pi\) cm\(^2\)
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    D: \(25\pi\) cm\(^2\)

    \(\pi r^2 = \pi \times 25 = 25\pi\).

  5. 5

    A circle has diameter 14 cm. What is its circumference, in terms of \(\pi\)?

    1. A\(7\pi\) cm
    2. B\(49\pi\) cm
    3. C\(14\pi\) cm
    4. D\(28\pi\) cm
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    C: \(14\pi\) cm

    \(C = \pi d = 14\pi\).

  6. 6

    A circle has diameter 10 cm. What is its area, in terms of \(\pi\)?

    1. A\(100\pi\) cm\(^2\)
    2. B\(25\pi\) cm\(^2\)
    3. C\(10\pi\) cm\(^2\)
    4. D\(50\pi\) cm\(^2\)
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    B: \(25\pi\) cm\(^2\)

    The radius is \(10 \div 2 = 5\), so \(A = \pi \times 5^2 = 25\pi\). Using 10 as the radius gives \(100\pi\).

  7. 7

    A rectangle measures 12 cm by 7 cm. A rectangle 5 cm by 3 cm is cut from one corner. What is the area of the remaining shape?

    1. A69 cm\(^2\)
    2. B99 cm\(^2\)
    3. C15 cm\(^2\)
    4. D84 cm\(^2\)
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    A: 69 cm\(^2\)

    \(12 \times 7 = 84\) and \(5 \times 3 = 15\), so \(84 - 15 = 69\).

  8. 8

    A sector has radius 9 cm and angle \(80^\circ\). What is its area, in terms of \(\pi\)?

    1. A\(4\pi\) cm\(^2\)
    2. B\(36\pi\) cm\(^2\)
    3. C\(81\pi\) cm\(^2\)
    4. D\(18\pi\) cm\(^2\)
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    D: \(18\pi\) cm\(^2\)

    \(\dfrac{80}{360} \times \pi \times 81 = \dfrac{2}{9} \times 81\pi = 18\pi\).

  9. 9

    A sector has radius 6 cm and angle \(60^\circ\). What is the length of its arc, in terms of \(\pi\)?

    1. A\(6\pi\) cm
    2. B\(\pi\) cm
    3. C\(2\pi\) cm
    4. D\(12\pi\) cm
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    C: \(2\pi\) cm

    \(\dfrac{60}{360} \times 2 \times \pi \times 6 = \dfrac{1}{6} \times 12\pi = 2\pi\).