Arc Length
The arc length of a circle depends on the radius and the angle at the centre.
Key idea
For a circle of radius r and a central angle θ measured in radians, the length of the arc is:
s = rθ
If the angle is given in degrees, first convert to radians: θ (radians) = θ (degrees) × π / 180.
Interactive Arc Length Lab
Drag the red point, or use the sliders, to explore how arc length changes.
Drag the red point on the circle. When close to special angles, it gently snaps (π/6, π/4, π/3, π/2, etc.). A right-angle marker appears at 90°.
Radius:
Angle: °
Angle in radians:
Arc length s = rθ:
Working (radians):
θ (radians) = θ° × π / 180
θ = × π / 180 =
s = rθ = × =
Working (degrees):
s = (θ° / 360) × 2πr
s = ( / 360) × 2π ×
=
≈
Find the arc length
A circle has radius 7 cm. Find the length of the arc that subtends an angle of 120° at the centre.
Given:
r = 7 cm
θ = 120°
Step-by-step
- Convert the angle to radians:
θ = 120° × π / 180 = (2π/3) radians. - Use the formula s = rθ:
s = 7 × (2π/3) = 14π/3 cm. - Approximate if needed:
s ≈ 14π/3 ≈ 14.66 cm (to 2 d.p.).
The arc length is 14π/3 cm (≈ 14.66 cm).
Try it yourself
Click “New question” to generate a random radius and angle. Work it out, then reveal the answer.
Radius: units
Angle: °
© Alexander Forrest