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 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.
- 2.A sector of a circle has radius 12 cm and angle 45°. Using π = 3.14, work out the area of the sector.
- 3.A sector of a circle has radius 6 cm and an area of 4.71 cm². Using π = 3.14, work out the angle of the sector.
- 4.A sector of a circle has radius 4 cm and angle 90°. Using π = 3.14, work out the area of the sector.
- 5.A sector of a circle has radius 10 cm and angle 150°. Using π = 3.14, work out the perimeter of the sector, to 1 decimal place.
- 6.A sector of a circle has radius 9 cm and angle 120°. Using π = 3.14, work out the arc length of the sector, to 1 decimal place.
- 7.A sector of a circle has radius 15 cm and angle 216°. Using π = 3.14, work out the arc length of the sector.
- 8.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.
- 9.A sector of a circle has radius 8 cm and takes up three-quarters of the full circle. Work out the perimeter of the sector, in terms of π.
- 10.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?
- 11.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.
- 12.A fan-shaped rug is in the shape of a sector of a circle with radius 4 m and angle 180°. It costs £4 per square metre to make. Using π = 3.14, work out the total cost of the rug.
- 13.A sector of a circle has radius 6 cm and angle 60°. Work out the arc length of the sector, in terms of π.
- 14.A goat is tied by a rope 7 m long to a post at a corner of a rectangular field, where two fences meet at a right angle. The goat can reach anywhere inside the field that the rope allows. Using π = 3.14, work out the area the goat can graze, to the nearest square metre.
- 15.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.
Answer key
- (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° — Sector area = (angle ÷ 360) × π × r², so 4.71 = (angle ÷ 360) × 3.14 × 36 = (angle ÷ 360) × 113.04. Dividing gives angle ÷ 360 = 4.71 ÷ 113.04 = 1/24, so angle = 360 ÷ 24 = 15°. (90° comes from forgetting to square the radius, using (angle ÷ 360) × 3.14 × 6 = 18.84 in place of 113.04; 3.75° comes from using the diameter, 12 cm, in place of the radius, giving (angle ÷ 360) × 3.14 × 144 = 452.16; 45° comes from using the arc length formula, (angle ÷ 360) × 2 × 3.14 × 6 = 37.68, instead of the sector area formula.)
- (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.)
- (d) 46.2 cm — The perimeter of a sector is the arc length plus its two straight radii. The circumference is 2 × 3.14 × 10 = 62.8 cm, and the arc is 150 ÷ 360 of that: 62.8 × 150 ÷ 360 = 26.2 cm (1 d.p.). Adding the two radii, 26.2 + 10 + 10 = 46.2 cm. Giving just the arc length, without adding the straight edges, gives 26.2 cm. Adding only ONE radius instead of two gives 36.2 cm. Using 150 ÷ 180 instead of 150 ÷ 360 for the fraction gives an arc of 52.3 cm and a perimeter of 72.3 cm.
- (d) 18.8 cm — Arc length is a fraction of the full circumference. Here 120 ÷ 360 = 1/3, and the full circumference is 2 × 3.14 × 9 = 56.52 cm, so the arc is 56.52 ÷ 3 = 18.84 cm, which rounds to 18.8 cm. Forgetting the fraction and giving the full circumference rounds to 56.5 cm. Using 9 cm as a diameter instead of a radius gives an arc of 9.4 cm. Using 120 ÷ 180 instead of 120 ÷ 360 for the fraction gives an arc of 37.7 cm.
- (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.)
- (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.)
- (a) 12π + 16 cm — A three-quarter sector's perimeter is the curved arc plus the two straight radii that close the shape. The full circumference is 2 × π × 8 = 16π cm, and three-quarters of that is 12π cm. Adding the two straight radii, 8 cm each, gives 12π + 16 cm. Leaving out the straight edges gives just 12π cm. Using one-quarter of the circumference, the piece left over rather than the piece asked for, gives 4π + 16 cm. Adding only one radius instead of two gives 12π + 8 cm.
- (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.
- (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°.
- (d) £100.48 — Area of the rug = (angle ÷ 360) × π × r² = (180 ÷ 360) × 3.14 × 16 = 0.5 × 50.24 = 25.12 m². Cost = 25.12 × £4 = £100.48. (£200.96 comes from finding the area of a full circle, 3.14 × 16 = 50.24 m², and forgetting the angle fraction before costing it; £25.12 comes from forgetting to square the radius, using 0.5 × 3.14 × 4 = 6.28 m², and then costing that; £50.24 comes from using the arc length formula, 0.5 × 2 × 3.14 × 4 = 12.56, in place of the area, and costing that.)
- (d) 2π cm — Arc length is the fraction θ/360 of the full circumference, 2πr. Substitute θ = 60 and r = 6: 60 out of 360 is one sixth, and the full circumference is 2π × 6 = 12π. One sixth of 12π is 2π, so the arc length is 2π cm. Choosing 12π cm uses the full circumference without scaling it down by the fraction θ/360 first. Choosing 6π cm applies the SECTOR AREA formula, (θ/360) × πr², instead of the arc length formula — that calculation actually gives 6π, which is the area in cm², not a length. Choosing π cm uses πr instead of 2πr, missing the factor of 2 in the circumference formula.
- (d) 38 m² — The two fences meet at a right angle, so the rope sweeps a quarter of a circle: 90 ÷ 360 = 1/4. Grazing area = (90 ÷ 360) × 3.14 × 7² = 0.25 × 153.86 = 38.465 m², which rounds to 38 m². (5 m² comes from forgetting to square the rope length; 154 m² comes from finding the area of a full circle and forgetting the angle fraction; 11 m² comes from using the arc length formula instead of the sector area formula.)
- (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.)
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