Area of a sector and arc length, with worked examples
A sector is a slice of a circle, like a slice of pizza, and its arc is the curved edge. Both are fractions of the whole circle: the angle at the centre tells you which fraction. This guide covers the formulas in degrees for GCSE and in radians for A level.
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The formulas in degrees (GCSE)
For a sector with radius r and angle θ degrees at the centre:
- Arc length = (θ/360) × 2πr
- Area of a sector = (θ/360) × πr²
- Perimeter of a sector = arc length + 2r
The fraction θ/360 is the share of the full circle. A 90° sector is a quarter of the circle, so its arc is a quarter of the circumference and its area a quarter of the area.
The formulas in radians (A level)
When θ is in radians, the formulas become simpler, because a full turn is 2π radians:
- Arc length s = rθ
- Area of a sector A = (1/2)r²θ
Use these only when the angle is in radians. To convert, multiply degrees by π/180. Our A level trigonometry guide covers radians in more depth.
Worked examples
Example 1: arc length and area
A sector has radius 8 cm and angle 45°. Find its arc length and area.
- 45/360 = 1/8 of the circle.
- Arc = (1/8) × 2π × 8 = 2π ≈ 6.28 cm.
- Area = (1/8) × π × 64 = 8π ≈ 25.1 cm².
Example 2: perimeter of a sector
Find the perimeter of a sector with radius 5 cm and angle 72°.
- Arc = (72/360) × 2π × 5 = 2π ≈ 6.28 cm.
- Perimeter = arc + two radii = 6.28 + 10 = 16.28 cm.
Example 3: finding the angle
A sector of radius 6 cm has area 30 cm². Find its angle to 1 decimal place.
- (θ/360) × π × 36 = 30.
- θ = 30 × 360 ÷ (36π) ≈ 95.5°.
Example 4: finding the radius
A sector with angle 105° has arc length 11 cm. Find its radius.
- (105/360) × 2πr = 11.
- r = 11 × 360 ÷ (105 × 2π) ≈ 6.00 cm.
Example 5: radians
A sector has radius 10 cm and angle 1.2 radians. Find its arc length and area.
- s = rθ = 10 × 1.2 = 12 cm.
- A = (1/2)r²θ = (1/2) × 100 × 1.2 = 60 cm².
Example 6: area of a segment
Find the area of the segment cut off by a chord, in a circle of radius 6 cm, where the angle at the centre is 60°.
- A segment is the sector minus the triangle.
- Sector = (60/360) × π × 36 = 6π ≈ 18.85 cm².
- Triangle = (1/2) × 6 × 6 × sin 60° ≈ 15.59 cm².
- Segment ≈ 18.85 − 15.59 = 3.26 cm².
Two exam-style problems
Example 7: a clock
The minute hand of a clock is 9 cm long. How far does its tip travel in 20 minutes, and what area does the hand sweep?
- 20 minutes is a third of an hour, so the angle is 360 ÷ 3 = 120°.
- Distance = (1/3) × 2π × 9 = 6π ≈ 18.85 cm.
- Area = (1/3) × π × 81 = 27π ≈ 84.8 cm².
Example 8: converting to radians first
A sector has radius 7 cm and angle 150°. Use radians to find its arc length and area.
- 150° × π/180 = 5π/6 ≈ 2.618 radians.
- s = rθ = 7 × 5π/6 ≈ 18.33 cm.
- A = (1/2) × 49 × 5π/6 ≈ 64.1 cm². The degree formulas give the same answers.
Sector, segment and arc: which is which?
| Part | What it is | How to find it |
|---|---|---|
| Arc | Part of the circumference | (θ/360) × 2πr, or rθ in radians |
| Sector | The region between two radii and an arc | (θ/360) × πr², or (1/2)r²θ in radians |
| Chord | A straight line joining two points on the circle | Trigonometry or the cosine rule |
| Segment | The region between a chord and an arc | Sector area minus triangle area |
For the triangle in a segment question, use area = (1/2)ab sin C, as in Example 6. Our sine and cosine rules guide covers it.
Practice questions
Try each one before opening the answer.
Q1. A sector has radius 12 cm and angle 30°. Find its arc length to 2 decimal places.
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Q2. Find the area of a sector with radius 9 cm and angle 80°, to 1 decimal place.
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Q3. Find the perimeter of a quarter circle (a 90° sector) of radius 4 cm.
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Q4. A sector has radius 5 cm and angle 0.8 radians. Find its arc length and area.
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Q5. A sector of radius 5 cm has arc length 7.5 cm. Find the angle in degrees, to 1 decimal place.
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Common mistakes
- Using the radian formulas with an angle in degrees.
- Giving the arc length when the question asks for the perimeter: add the two radii.
- Using the diameter instead of the radius.
- Forgetting to subtract the triangle when finding a segment.
Frequently asked questions
What is the formula for the area of a sector?
(θ/360) × πr² with θ in degrees, or (1/2)r²θ with θ in radians.
What is the formula for arc length?
(θ/360) × 2πr with θ in degrees, or rθ with θ in radians.
How do you find the perimeter of a sector?
Add the arc length and two radii.
Are these formulas on the GCSE formula sheet?
For GCSE Maths exams in 2025, 2026 and 2027 a formulae sheet is provided. Understand where the formulas come from, as a fraction of the circle, so you can use them quickly.
Key takeaways
- Arc = (θ/360) × 2πr; sector area = (θ/360) × πr².
- In radians: s = rθ and A = (1/2)r²θ.
- Perimeter of a sector = arc + 2r; segment = sector − triangle.