Area, Perimeter and Circles

1. 1 Calculate [5 marks]

   The diagram shows a sector of a circle with radius 6 cm and angle \(120^\circ\). Give your answers in terms of \(\pi\). (a) Calculate the area of the sector. [2 marks] (b) Calculate the length of the arc. [2 marks] (c) Write down the perimeter of the sector. [1 mark]

   

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   Model answer

   \(\dfrac{120}{360} = \dfrac{1}{3}\). (a) The area of the circle is \(\pi \times 36 = 36\pi\), so the sector is \(12\pi\) cm\(^2\). (b) The circumference is \(2\pi \times 6 = 12\pi\), so the arc is \(4\pi\) cm. (c) The perimeter is the arc plus two radii, \(4\pi + 6 + 6 = 12 + 4\pi\) cm.

   Mark scheme

   • (a) \(\dfrac{1}{3} \times \pi \times 6^2\) — M1
   • (a) \(12\pi\) — A1
   • (b) \(\dfrac{1}{3} \times 2 \times \pi \times 6\) — M1
   • (b) \(4\pi\) — A1
   • (c) \(12 + 4\pi\), with their arc if correct — B1
2. 2 Calculate [2 marks]

   A triangle has a base of 15 cm and a perpendicular height of 8 cm. Calculate the area of the triangle.

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   Model answer

   \(\dfrac{1}{2} \times 15 \times 8 = 60\) cm\(^2\).

   Mark scheme

   • \(\dfrac{1}{2} \times 15 \times 8\) or \(15 \times 8 \div 2\) — M1
   • 60 cm\(^2\) — A1
3. 3 Calculate [3 marks]

   A trapezium has an area of 72 cm\(^2\). Its parallel sides are 7 cm and 11 cm long. Calculate the distance between the parallel sides.

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   Model answer

   \(\dfrac{1}{2}(7 + 11) \times h = 72\), so \(9h = 72\) and \(h = 8\) cm.

   Mark scheme

   • \(\dfrac{1}{2}(7 + 11) \times h = 72\) or \(9h = 72\) — M1
   • \(72 \div 9\) — M1
   • 8 cm — A1
4. 4 Calculate [3 marks]

   A circle has diameter 16 cm. Give your answers in terms of \(\pi\). (a) Calculate the circumference of the circle. [1 mark] (b) Calculate the area of the circle. [2 marks]

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   Model answer

   (a) \(\pi \times 16 = 16\pi\) cm. (b) The radius is 8 cm, so the area is \(\pi \times 8^2 = 64\pi\) cm\(^2\).

   Mark scheme

   • (a) \(16\pi\) — B1
   • (b) \(\pi \times 8^2\) — M1
   • (b) \(64\pi\) — A1
5. 5 Calculate [4 marks]

   A semicircle has radius 5 cm. Give your answers in terms of \(\pi\). (a) Calculate the area of the semicircle. [2 marks] (b) Calculate the perimeter of the semicircle. [2 marks]

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   Model answer

   (a) \(\dfrac{1}{2} \times \pi \times 5^2 = 12.5\pi\) cm\(^2\). (b) The curved edge is \(\dfrac{1}{2} \times \pi \times 10 = 5\pi\) and the straight edge is 10 cm, so the perimeter is \(10 + 5\pi\) cm.

   Mark scheme

   • (a) \(\dfrac{1}{2} \times \pi \times 5^2\) — M1
   • (a) \(12.5\pi\) or \(\dfrac{25\pi}{2}\) — A1
   • (b) \(5\pi\) or the diameter 10 found — M1
   • (b) \(10 + 5\pi\) — A1
6. 6 Calculate [3 marks]

   A circular lawn has a radius of 8 m. A path 2 m wide goes all the way round the lawn. Calculate the area of the path. Give your answer in terms of \(\pi\).

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   Model answer

   The lawn and path together have radius 10 m, so their area is \(100\pi\). The lawn has area \(64\pi\). The path is \(100\pi - 64\pi = 36\pi\) m\(^2\).

   Mark scheme

   • \(\pi \times 10^2\) or \(\pi \times 8^2\) — M1
   • \(100\pi - 64\pi\) — M1
   • \(36\pi\) m\(^2\) — 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

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