All subjects · Mathematics · WMA11 (P1)

Radians, arc length and sector area

Radians are the natural unit for angles in A level maths. With radians, the formulas for arcs and sectors become very simple.

Radians

π radians = 180°. So 1 radian is about 57.3°.

degrees to radians: multiply by π/180
radians to degrees: multiply by 180/π

Useful values: 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6.

Arc and sector

For a circle of radius r and angle θ in radians:

arc length s = rθ
sector area A = (1/2) r² θ

Segments and perimeters

A segment is a sector minus the triangle: area = (1/2) r² θ - (1/2) r² sin θ = (1/2) r² (θ - sin θ). The perimeter of a sector is 2r + rθ. Set your calculator to radians.

Worked example

A sector has radius 6 cm and angle 1.2 radians. Find its arc length and area.

s = rθ = 6 x 1.2 = 7.2
A = 0.5 x 36 x 1.2 = 21.6

Answer: Arc 7.2 cm, area 21.6 cm²

Key idea: π rad = 180°; s = rθ; A = (1/2) r² θ; segment = (1/2) r² (θ - sin θ); θ must be in radians.

Check yourself

Write 135° in radians.
- 2π/3
- 5π/6
- 3π/2
- 3π/4
Show answer
3π/4. 135 x π/180 = 3π/4.
Write π/5 radians in degrees.
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36 °. π/5 x 180/π = 36°.
Find the arc length of a sector with radius 8 cm and angle 0.75 radians.
Show answer
6 cm. s = rθ = 8 x 0.75 = 6 cm.
Find the area of a sector with radius 10 cm and angle 0.6 radians.
Show answer
30 cm². A = 0.5 x 100 x 0.6 = 30 cm².
A sector of radius 6 cm has area 27 cm². Find the angle in radians.
Show answer
1.5. 27 = 0.5 x 36 x θ = 18θ, so θ = 1.5.
Find the perimeter of a sector with radius 5 cm and angle 1.1 radians.
Show answer
15.5 cm. Perimeter = 2r + rθ = 10 + 5.5 = 15.5 cm.
Find the area of the segment cut off by a chord in a circle of radius 5 cm, where the angle at the centre is π/3. Give your answer to 2 decimal places.
Show answer
2.26 cm². Area = 0.5 x 25 x (π/3 - sin(π/3)) = 2.26 cm².

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.