Calculate the length of the arc of a circle if:
- (i) the radius is 3.5 cm and the angle at the centre is 60°, and
- (ii) the radius is 6.3 m and the angle at the centre is 120°.
Answer: (i) 113 cm ≈ 3.67 cm (ii) 13.2 m
Step-by-step solution
Idea: An arc that subtends θ° at the centre is θ360 of the whole circumference: l = 2πr × θ360°.
(i) the radius is 3.5 cm and the angle at the centre is 60°, and
- Arc length = 2πr × θ360° = 2 × 227 × 3.5 × 60°360°.½ mark
- = 22 × 16 = 113 ≈ 3.67 cm.1 mark
113 cm ≈ 3.67 cm
(ii) the radius is 6.3 m and the angle at the centre is 120°.
- Arc length = 2 × 227 × 6.3 × 120°360°.½ mark
- 2 × 227 × 6.3 = 39.6, and 39.6 × 13 = 13.2 m.1 mark
13.2 m
(i) 11/3 cm ≈ 3.67 cm (ii) 13.2 m
Check: (i) 60° is 16 of a turn and the full circle is 22 cm, so 22 ÷ 6 ≈ 3.67 ✓. (ii) 120° is 13 of a turn and the full circle is 39.6 m, so 39.6 ÷ 3 = 13.2 ✓.
Answer to write in the exam
(i)
l = 2πr × θ360° = 2 × 227 × 3.5 × 60°360°
= 22 × 16
∴ l = 113 cm ≈ 3.67 cm
(ii)
l = 2 × 227 × 6.3 × 120°360°
= 39.6 × 13
∴ l = 13.2 m
Common mistakes that cost marks
- Using πr2 × θ360, which is the area of a sector, not the arc length.
- Using πr instead of 2πr before taking the fraction.
- Forgetting the unit m in (ii).
How this can come in the exam
MCQ (1 mark)
The length of an arc of a circle of radius 14 cm subtending 45° at the centre is (use π = 227)
- 11 cm
- 22 cm
- 77 cm
- 88 cm
Show answer
(A) 11 cm
2 × 227 × 14 × 45360 = 88 × 18 = 11 cm.
Try one yourself
Find the length of an arc of a circle of radius 21 cm that subtends 30° at the centre. (Use π = 227.)
Show answer
2 × 227 × 21 × 30360 = 132 × 112 = 11 cm.
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