Arc lengths & sectors · Ages 14–15
Arc length
Arc = θ360 × 2πr.
How to do it
Method
Arc = θ/360 × 2πr
r = 6, θ = 60°: 12π ÷ 6 = 2π cm
Another way
Cancel first: θr/180
60 × 6 ÷ 180 = 2 → 2π
In full
A sector is part of a circle: an angle θ at the centre is θ360 of the whole turn, so its arc is θ360 of the circumference. Arc length = θ360 × 2πr. Radius 6 cm, angle 60°: 60360 = 16 and the circumference is 2π × 6 = 12π, so the arc is 12π ÷ 6 = 2π cm. Or cancel first, then multiply: θ360 × 2πr = θr180 × π = 60 × 6 ÷ 180 × π = 2π, the same. Keep the π, or put a value in: with π as 3.14, 6.28 cm; with the calculator's π, to 1 decimal place, 6.3 cm. The other way round, from the arc: θ = arc ÷ circumference × 360; the radius: r = arc × 180 ÷ (θπ).
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The sector in the picture has a radius of 6 cm and an angle of 30° at the centre. What is its arc length, in terms of π?
Show the answer and working
Answer: 1 π cm
Arc = 30360 × 2π × 6 = 30360 × 12π = 1π cm.
Example 2
A sector of radius 5 cm and angle 144° has an arc length of 4π cm.
True or false?
Show the answer and working
Answer: True
Arc = 144360 × 2π × 5 = 4π cm: the circumference is 2πr, not πr nor 4πr.
Example 3
A clock's minute hand is 13 cm long. Turning from 12 to the 30-minute mark, it turns °, and its tip travels π cm.
Fill in each blank
Show the answer and working
Answer: 180, 13
60 minutes make 360°, 6° a minute: 30 × 6 = 180°; the arc is 180360 × 2π × 13 = 13π cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The sector in the picture has a radius of 5 cm and an angle of 36° at the centre. What is its arc length, in terms of π?
Answer:
2.The sector in the picture has a radius of 8 cm and an arc of 6π cm. What is its angle at the centre, in degrees?
Answer:
3.The sector in the picture has a radius of 12 cm and an angle of 35°. Correct to 1 decimal place, what is its arc length, in cm?
Answer:
4.A clock's minute hand is 15 cm long. Turning from 12 to the 30-minute mark, it turns °, and its tip travels π cm.
Fill in each blank
Answer:
5.The sector in the picture has an angle of 150° and an arc of 7.5π cm. What is its radius, in cm?
Answer:
All the answers
- 1. 1 π cm
- 2. 135°
- 3. 7.3 cm
- 4. 180, 15
- 5. 9 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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