Printable · GCSE Foundation · ages 14-16
Arc lengths and sector areas worksheet — GCSE Foundation
Fifteen questions on "arc lengths and sector areas" — DfE statement G18. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Arc lengths and sector areas worksheet — GCSE Foundation
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- 1.A circular decoration has a radius of 10 cm. A ribbon costs £0.50 per centimetre and is fixed along the curved edge (the arc) of a sector-shaped section with an angle of 90°. Using π = 3.14, work out the cost of the ribbon for this section.
- 2.A sector of a circle has radius 7 cm and angle 90°. Using π = 3.14, work out the arc length of the sector, to 1 decimal place.
- 3.A car's windscreen wiper blade is 40 cm long and sweeps through an angle of 110° as it wipes the glass. Using π = 3.14, work out the area of glass it wipes, to the nearest cm².
- 4.A sector of a circle has radius 12 cm and angle 45°. Using π = 3.14, work out the area of the sector.
- 5.A sector of a circle has angle 120° and an arc length of 31.4 cm. Using π = 3.14, work out the radius of the circle.
- 6.A running track has a straight section and a semicircular bend. The bend is a semicircle with radius 32 m. Work out the length of the curved part of the bend, to 1 decimal place. (Use π = 3.14.)
- 7.A sector of a circle has radius 10 cm and angle 72°. Using π = 3.14, work out the arc length of the sector.
- 8.A ship's radar can detect other ships up to 20 km away, over a sector-shaped area with angle 90°. A monitoring company charges £0.50 per square kilometre covered by one radar sweep. Using π = 3.14, work out the total charge for one sweep.
- 9.Priya says: 'To find the arc length of a sector, you divide the circumference by the angle in degrees.' Which statement about Priya's method is correct?
- 10.A sector of a circle has radius 6 cm and angle 150°. Using π = 3.14, work out the perimeter of the sector.
- 11.A sector of a circle has an angle of 150°. Work out what fraction of the circle's area the sector is.
- 12.A sector of a circle has radius 15 cm and angle 216°. Using π = 3.14, work out the arc length of the sector.
- 13.A sector has an angle of 45°. Priti says: 'The area of the sector is one eighth of the area of the full circle, whatever the radius is.' Is Priti correct?
- 14.A sector of a circle has radius 4 cm and angle 90°. Using π = 3.14, work out the area of the sector.
- 15.A sector of a circle has radius 6 cm and an arc length of 15.7 cm. Using π = 3.14, work out the angle of the sector.
Answer key
- (c) £7.85 — Arc length = (90 ÷ 360) × 2 × 3.14 × 10 = 0.25 × 62.8 = 15.7 cm. Cost = 15.7 × £0.50 = £7.85. (£31.40 comes from finding the full circumference and forgetting the angle fraction; £15.70 comes from using the diameter, 20 cm, in place of the radius; £157.00 comes from multiplying the arc length by the radius instead of by the cost per centimetre.)
- (c) 11.0 cm — Arc length = (angle ÷ 360) × 2 × π × r = (90 ÷ 360) × 2 × 3.14 × 7 = 0.25 × 43.96 = 10.99 cm, which rounds to 11.0 cm. (44.0 cm comes from finding the full circumference, 2 × 3.14 × 7 = 43.96 cm, and forgetting the angle fraction; 5.5 cm comes from leaving out the factor of 2, using π × r instead of 2 × π × r, giving 0.25 × 3.14 × 7 = 5.495 cm; 22.0 cm comes from using the diameter, 14 cm, in place of the radius, giving 0.25 × 2 × 3.14 × 14 = 21.98 cm.)
- (d) 1535 cm² — The wiper sweeps out a sector of radius 40 cm, the blade length, through an angle of 110°. Sector area is angle ÷ 360 × π × radius²: 110 ÷ 360 × 3.14 × 1600 = 1535.1 cm², which rounds to 1535 cm². Forgetting to square the radius, using radius instead of radius², gives 38 cm². Using 110 ÷ 180 instead of 110 ÷ 360 for the fraction gives 3070 cm². Treating the 40 cm blade length as a diameter, so using a radius of 20 cm, gives 384 cm².
- (c) 56.52 cm² — Sector area = (45 ÷ 360) × 3.14 × 12² = 0.125 × 3.14 × 144 = 56.52 cm². (4.71 cm² comes from forgetting to square the radius; 452.16 cm² comes from finding the area of the whole circle and forgetting the angle fraction; 9.42 cm² comes from using the arc length formula instead of the sector area formula.)
- (d) 15 cm — Arc length = (angle ÷ 360) × 2 × π × r. Here 120 ÷ 360 = 1/3, and 2 × 3.14 = 6.28, so 31.4 = (1/3) × 6.28 × r. Multiplying both sides by 3 gives 6.28 × r = 94.2, so r = 94.2 ÷ 6.28 = 15 cm. (5 cm comes from forgetting the angle fraction altogether and dividing the arc length by 2 × π alone: 31.4 ÷ 6.28 = 5; 30 cm comes from leaving out the factor of 2, dividing by (1/3) × 3.14 = 1.0467 instead of (1/3) × 6.28: 31.4 ÷ 1.0467 = 30; 7.5 cm comes from correctly finding a radius of 15 cm but then treating that 15 cm as a diameter and halving it.)
- (c) 100.5 m — The curved part of a semicircular bend is half of a full circle's circumference. The full circumference would be 2 × 3.14 × 32 = 200.96 m, and half of that is 200.96 ÷ 2 = 100.48 m, which rounds to 100.5 m. Choosing 201.0 m uses the FULL circumference, forgetting to halve it for a semicircle. Choosing 50.2 m halves the radius as well as taking a semicircle, using 3.14 × 16 = 50.24 m instead of the correct radius of 32 m. Choosing 64.0 m uses the diameter, 2 × 32 = 64, as if it were the curved length, ignoring π and the semicircle shape entirely.
- (d) 12.56 cm — Arc length = (angle ÷ 360) × 2 × π × r = (72 ÷ 360) × 2 × 3.14 × 10 = 0.2 × 62.8 = 12.56 cm. (62.8 cm comes from finding the full circumference and forgetting to take the angle fraction; 6.28 cm comes from leaving out the factor of 2, using πr instead of 2πr; 25.12 cm comes from using the diameter, 20 cm, in place of the radius.)
- (a) £157.00 — Area covered = (angle ÷ 360) × π × r² = (90 ÷ 360) × 3.14 × 400 = 0.25 × 1256 = 314 km². Charge = 314 × £0.50 = £157.00. (£7.85 comes from forgetting to square the radius, using 0.25 × 3.14 × 20 = 15.7 km² and then charging that; £628.00 comes from finding the area of a full circle, 3.14 × 400 = 1256 km², and forgetting the angle fraction before charging; £15.70 comes from using the arc length formula, 0.25 × 2 × 3.14 × 20 = 31.4, in place of the area, and charging that.)
- (a) Wrong: multiply by angle ÷ 360, not divide by it. — The arc length is the fraction angle ÷ 360 of the circumference, and that fraction is MULTIPLIED by the circumference — it is never divided by the angle itself. So Priya's method, dividing by the angle, is wrong, and the correct statement names both the fraction and the multiplication. Saying she is correct accepts her wrong division. Saying the arc length is the full circumference multiplied by the angle in degrees, with no division at all, drops the 360 that turns the angle into a fraction. Saying the arc length is the radius multiplied by the angle in degrees swaps the circumference for the radius, which is a different and, in degrees, incorrect formula.
- (c) 27.7 cm — Arc length = (150 ÷ 360) × 2 × 3.14 × 6 = (5 ÷ 12) × 37.68 = 15.7 cm. The perimeter of a sector also includes the two straight radii, so perimeter = 15.7 + 6 + 6 = 27.7 cm. (15.7 cm comes from stopping after the arc length and forgetting the two straight edges; 21.7 cm comes from adding only one radius instead of two; 31.4 cm comes from doubling the arc length instead of adding the two straight edges.)
- (a) 5/12 — A sector's area is always the same fraction of the circle as its angle is of the full 360° turn. Here that fraction is 150 out of 360, which simplifies (divide both numbers by 30) to 5/12. Treating a full turn as 180° instead of 360° gives 5/6. Confusing the 150° given with the 180° of a half turn gives 1/2. Working out the fraction OUTSIDE the sector instead of inside it gives 7/12.
- (b) 56.52 cm — Arc length = (216 ÷ 360) × 2 × 3.14 × 15 = 0.6 × 94.2 = 56.52 cm. (28.26 cm comes from leaving out the factor of 2, using πr instead of 2πr; 94.2 cm comes from finding the full circumference and forgetting the angle fraction; 113.04 cm comes from using the diameter, 30 cm, instead of the radius.)
- (d) Yes, because 45 ÷ 360 = 1/8, and this fraction applies to any radius. — Priti is correct: 45 ÷ 360 = 1/8, and because sector area = (angle ÷ 360) × full circle area, this fraction depends only on the angle, not on the radius, so it holds for every circle. (The statement giving 45 ÷ 360 = 1/6 comes from an arithmetic slip when simplifying the fraction; the statement that the fraction depends on the radius is wrong because the radius cancels out of the angle fraction in the sector-area formula; the statement adding that it only works for this particular radius is wrong because the 1/8 fraction is true for every radius, not a special case.)
- (b) 12.56 cm² — Sector area = (angle ÷ 360) × π × r² = (90 ÷ 360) × 3.14 × 4² = 0.25 × 3.14 × 16 = 12.56 cm². (3.14 cm² comes from forgetting to square the radius; 50.24 cm² comes from finding the area of the whole circle and forgetting the angle fraction; 6.28 cm² comes from using the arc length formula instead of the sector area formula.)
- (a) 150° — The arc length is the same fraction of the circumference as the angle is of 360°. The full circumference is 2 × 3.14 × 6 = 37.68 cm, so the angle is 15.7 ÷ 37.68 × 360 = 150°. Treating the 6 cm as a diameter instead of a radius gives a circumference of 18.84 cm and an angle of 300°. Using π = 3 instead of the given 3.14 gives a circumference of 36 cm and an angle of 157°. Treating the fraction 15.7 ÷ 37.68 as being out of 100 rather than 360 gives about 42°.
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