Area of a Part of a Circle

14 Grade 10 Core Mathematics questions on Area of a Part of a Circle, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   1 mark

   A circular flower bed of radius 7 m in a school compound is surrounded by a paved path of uniform width 1 m. Taking $\pi =\frac{22}{7}$, the area of the path alone is:

   A. 201.1 m²

   B. 154.0 m²

   C. 47.1 m²

   D. 3.1 m²

   (1 mark)

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2. Question 2

   5 marks

   A rotating sprinkler on an irrigated rice scheme in Kirinyaga throws water a distance of 12 m and is set to turn through 150°. Taking π = 3.142, calculate (a) the area watered at this setting, (b) the extra area watered when the setting is changed to 210°. (c) Suggest one reason why the area actually wetted may be smaller than the value calculated.

   (5 marks)

   Marking scheme

   Award 1 mark per point, maximum 5. (a) 1. Area = $\frac{150}{360}\times 3.142\times 12^2$. ✓ 2. = 188.5 m². ✓ (b) 3. Area at 210° = $\frac{210}{360}\times 3.142\times 144=263.9m^2$. ✓ 4. Extra area = $263.9-188.5=75.4m^2$. ✓ (c) 5. Any one valid reason: wind blows part of the spray outside the sector; low water pressure shortens the throw; obstacles such as bunds, trees or crops block the jet; the sector may overlap ground already wetted. ✓

3. Question 3

   1 mark

   A school badge is cut from a square piece of card of side 14 cm by removing a quarter circle of radius 14 cm centred at one corner of the card. Taking $\pi =\frac{22}{7}$, the area of card that remains is:

   A. 119 cm²

   B. 154 cm²

   C. 42 cm²

   D. 196 cm²

   (1 mark)

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4. Question 4

   1 mark

   A sector cut from a circular metal plate of radius 21 cm has an area of 231 cm². Taking $\pi =\frac{22}{7}$, the angle the sector subtends at the centre is:

   A. 90°

   B. 30°

   C. 120°

   D. 60°

   (1 mark)

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5. Question 5

   1 mark

   A circular tray of radius 14 cm is divided by two radii into a minor sector of angle 100° and a major sector. Taking $\pi =\frac{22}{7}$, the area of the major sector is:

   A. 308.0 cm²

   B. 616.0 cm²

   C. 171.1 cm²

   D. 444.9 cm²

   (1 mark)

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6. Question 6

   1 mark

   A chord of a circle of radius 10 cm subtends an angle of 90° at the centre. Taking π = 3.142, the area of the minor segment cut off by the chord is:

   A. 128.6 cm²

   B. 50.0 cm²

   C. 78.6 cm²

   D. 28.6 cm²

   (1 mark)

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7. Question 7

   2 marks

   Distinguish between the arc length of a sector and the area of the same sector.

   (2 marks)

   Marking scheme

   Award 1 mark per point, maximum 2. 1. Arc length is the part of the circumference lying between the two radii — a one-dimensional length measured in cm or m, found from $\theta /360\times 2\pi r$. ✓ 2. Area of the sector is the region enclosed by the two radii and the arc — a two-dimensional space measured in cm² or m², found from $\theta /360\times \pi$r². ✓ Accept the answer if the distinction of dimension/units is made and both formulae or descriptions are correct.

8. Question 8

   4 marks

   A sector of a circle of radius 21 cm subtends an angle of 120° at the centre. The chord joining the two ends of the arc is 36.4 cm long, and the perpendicular distance from the centre of the circle to this chord is 10.5 cm. Taking $\pi =\frac{22}{7}$, calculate (a) the area of the sector, (b) the area of the minor segment cut off by the chord.

   (4 marks)

   Marking scheme

   Award 1 mark per point, maximum 4. (a) 1. Area of whole circle = $\frac{22}{7}\times 21^2=1386$ cm², and the sector is $\frac{120}{360}=\frac{1}{3}$ of it. ✓ 2. Area of sector = $\frac{1}{3}\times 1386=462$ cm². ✓ (b) 3. Area of the triangle formed by the two radii and the chord = $\frac{1}{2}\times 36.4\times 10.5=191.1$ cm². ✓ 4. Area of segment = area of sector − area of triangle = 462 − 191.1 = 270.9 cm². ✓ Allow 270.9 cm² or 271 cm² from correct working.

9. Question 9

   5 marks

   A carpenter in Kisumu makes a table top from a rectangular board 2 m long and 1.4 m wide, and fixes a semicircular piece of diameter 1.4 m to each of the two shorter ends. Taking $\pi =\frac{22}{7}$, calculate (a) the area of the rectangular part, (b) the total area of the two semicircular pieces, (c) the total area of the table top, (d) the cost of varnishing the top surface at KSh 450 per square metre.

   (5 marks)

   Marking scheme

   Award 1 mark per point, maximum 5. (a) 1. Rectangular part = $2\times 1.4=2.8m^2$. ✓ (b) 2. The two semicircles of radius 0.7 m together make one full circle = $\frac{22}{7}\times 0.7^2=1.54m^2$. ✓ (c) 3. Total area = $2.8+1.54=4.34m^2$. ✓ (d) 4. Cost = $4.34\times 450$. ✓ 5. = KSh 1 953. ✓ A candidate who treats the two semicircles as two full circles forfeits marks 2, 3 and 5.

10. Question 10

    3 marks

    A sector subtending 90° at the centre of a circle has an area of 154 cm². Taking $\pi =\frac{22}{7}$, calculate the radius of the circle.

    (3 marks)

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11. Question 11

    1 mark

    A sector of a circle of radius 8 cm subtends an angle of 135° at the centre. Taking π = 3.142, the area of the sector is:

    A. 150.8 cm²

    B. 75.4 cm²

    C. 18.8 cm²

    D. 201.1 cm²

    (1 mark)

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12. Question 12

    1 mark

    A classroom window has a rectangular pane 1.4 m wide and 1 m high, with a semicircular pane of diameter 1.4 m fitted directly above it. Taking $\pi =\frac{22}{7}$, the total area of glass in the window is:

    A. 2.94 m²

    B. 1.54 m²

    C. 2.17 m²

    D. 1.40 m²

    (1 mark)

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13. Question 13

    3 marks

    A semicircular vegetable nursery of radius 7 m is enlarged to a semicircle of radius 14 m. Taking $\pi =\frac{22}{7}$, explain why the area of the nursery becomes four times as large, and not twice as large.

    (3 marks)

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14. Question 14

    4 marks

    The circular floor of a cattle boma has a circumference of 44 m. Taking $\pi =\frac{22}{7}$, calculate (a) the radius of the floor, (b) the area of the three-quarter part of the floor that is roofed.

    (4 marks)

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