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Exam questions · Maths · Geometry and Measures

Area, Perimeter and Circles

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

    The diagram shows a rectangle with a semicircle attached to one side. Give your answers correct to 3 significant figures. (a) Work out the area of the shape. [3 marks] (b) Work out the perimeter of the shape. [2 marks]

    A rectangle 10 cm by 6 cm with a semicircle on one of its 6 cm sides.
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    Model answer

    (a) The rectangle has area \(10 \times 6 = 60\) cm\(^2\). The semicircle has radius 3 cm, so its area is \(\dfrac{1}{2} \times \pi \times 3^2 = 14.14\) cm\(^2\). The total is \(74.1\) cm\(^2\). (b) The curved edge is \(\dfrac{1}{2} \times \pi \times 6 = 9.425\), and the straight edges are \(10 + 10 + 6 = 26\). The perimeter is \(35.4\) cm.

    Mark scheme

    • (a) \(60\) found — M1
    • (a) \(\dfrac{1}{2} \times \pi \times 3^2\) — M1
    • (a) \(74.1\) — A1
    • (b) \(\dfrac{1}{2} \times \pi \times 6\) or \(26\) found — M1
    • (b) \(35.4\) — A1
  2. 2 Work out [2 marks]

    A parallelogram has a base of 12 cm, a slanted side of 8 cm and a perpendicular height of 7 cm. Work out the area of the parallelogram.

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    Model answer

    Area \(=\) base \(\times\) perpendicular height \(= 12 \times 7 = 84\) cm\(^2\). The slanted side is not needed.

    Mark scheme

    • \(12 \times 7\) — M1
    • 84 cm\(^2\) — A1
  3. 3 Work out [4 marks]

    A rectangle is 14 cm by 9 cm. A smaller rectangle measuring 6 cm by 4 cm is cut out of one corner. (a) Work out the area of the shape that is left. [2 marks] (b) Show that the perimeter of the shape is 46 cm. [2 marks]

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    Model answer

    (a) \(14 \times 9 - 6 \times 4 = 126 - 24 = 102\) cm\(^2\). (b) Cutting a rectangle out of a corner does not change the perimeter, because the two new edges replace two edges of the same total length: \(2 \times (14 + 9) = 46\) cm.

    Mark scheme

    • (a) \(14 \times 9 - 6 \times 4\) — M1
    • (a) 102 — A1
    • (b) \(14 + 9 + 14 + 9\) or the shape's edges added — M1
    • (b) 46 with a reason or full working — Q1
  4. 4 Work out [3 marks]

    The circumference of a circle is 62.8 cm. Work out the area of the circle. Give your answer correct to 3 significant figures.

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    Model answer

    \(2\pi r = 62.8\), so \(r = 9.995\) cm. The area is \(\pi \times 9.995^2 = 314\) cm\(^2\).

    Mark scheme

    • \(r = \dfrac{62.8}{2\pi}\) — M1
    • \(\pi \times 9.995^2\) — M1
    • \(314\) cm\(^2\) — A1
  5. 5 Work out [3 marks]

    A circular pond has a radius of 4 m. A path 1 m wide goes all the way round the pond. Work out the area of the path. Give your answer correct to 3 significant figures.

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    Model answer

    The pond and path together have radius 5 m, so their area is \(\pi \times 25\). The pond has area \(\pi \times 16\). The path is \(\pi \times 25 - \pi \times 16 = 28.3\) m\(^2\).

    Mark scheme

    • \(\pi \times 5^2\) or \(\pi \times 4^2\) — M1
    • \(\pi \times 5^2 - \pi \times 4^2\) — M1
    • \(28.3\) m\(^2\) — A1
  6. 6 Work out [4 marks]

    A sector of a circle has radius 10 cm and angle \(72^\circ\). Give your answers correct to 3 significant figures. (a) Work out the length of the arc. [2 marks] (b) Work out the perimeter of the sector. [2 marks]

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    Model answer

    (a) \(\dfrac{72}{360} \times 2 \times \pi \times 10 = 12.6\) cm. (b) The perimeter is the arc plus two radii: \(12.57 + 10 + 10 = 32.6\) cm.

    Mark scheme

    • (a) \(\dfrac{72}{360} \times 2 \times \pi \times 10\) — M1
    • (a) \(12.6\) — A1
    • (b) Arc \(+ 10 + 10\) — M1
    • (b) \(32.6\) — 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

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