Arc Length Calculator
Enter a radius and a central angle — we'll use π to compute the exact arc length along the circle.
Arc length formula
The arc length (s) is the distance travelled along the curve of a circle. It depends on two things: the circle's radius (r) and the central angle (θ) that the arc subtends at the center.
- Radians:
s = r · θ - Degrees:
s = r · θ · (π / 180)
When θ equals a full turn (360°, or 2π radians), the arc length equals the entire circumference of the circle, 2πr. That's the sanity check every arc length formula obeys.
Worked example
A pie slice cut from a 20 cm radius pie has a central angle of 45°. The crust along the outside of the slice measures:
s = 20 · 45 · (π/180) = 5π ≈ 15.7080 cm
Why radians are elegant
Radians are defined so that an angle of 1 radian subtends an arc equal to the radius. That's why the formula s = rθ is so clean — radians are essentially "how many radii fit along the arc." Degrees need an extra conversion factor of π/180 because degrees are arbitrary units of angle.