What is the Arc Length Calculator?
This tool computes the length of a circular arc — the curved distance along the edge of a circle — from the circle's radius and the central angle that the arc subtends. It works for any unit of length (cm, m, inches, feet) because the result simply takes on the same unit as the radius you enter. You can supply the angle in either radians or degrees.
How to use it
Enter the radius r of the circle and the central angle θ. Choose whether your angle is in radians or degrees, then read off the arc length. The result table also shows the equivalent angle in the other unit so you can sanity-check your input.
The formula explained
The core relationship is $$s = r\theta$$ where θ must be in radians. This comes from the definition of a radian: an angle of 1 radian sweeps out an arc exactly equal in length to the radius. A full circle is \(2\pi\) radians, giving the familiar circumference \(2\pi r\). If your angle is in degrees, first convert it with $$\theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180}$$ then multiply by the radius.
Worked example
Suppose a circle has a radius of 10 cm and a central angle of 90°. Convert the angle: $$90 \times \frac{\pi}{180} = \frac{\pi}{2} \approx 1.5708 \text{ radians}$$ Then $$s = 10 \times 1.5708 = 15.708 \text{ cm}$$ That is a quarter of the full circumference (\(2\pi \times 10 \approx 62.83\) cm), exactly as expected.
Key Terms
- Arc length (\(s\))
- The distance measured along the curved edge of a circle between two points. Calculated as \(s = r\theta\) when the central angle \(\theta\) is in radians.
- Radius (\(r\))
- The straight-line distance from the center of the circle to any point on its edge. The arc length scales directly with the radius.
- Central angle (\(\theta\))
- The angle formed at the center of the circle by the two radii that bound the arc. It must be in radians to use \(s = r\theta\) directly.
- Radian
- A unit of angle defined so that an angle of 1 radian subtends an arc equal in length to the radius. A full circle is \(2\pi\) radians \(\approx 6.2832\) rad \(= 360^\circ\).
- Central angle subtending an arc
- An arc is said to be subtended by its central angle when the angle's two sides (radii) meet the circle at the arc's endpoints. A larger subtended angle corresponds to a longer arc.
- Circumference (\(C\))
- The total distance around the circle, equal to the arc length of a complete \(360^\circ\) (\(2\pi\) rad) angle: \(C = 2\pi r\).
FAQ
What units does the arc length come out in? The same length unit as the radius — the formula is unit-agnostic.
Do I have to convert degrees myself? No. Just select "Degrees" and the calculator converts to radians internally.
Can I find the angle if I know the arc length? Yes, rearrange to \(\theta = s / r\) (in radians); this calculator solves for s, but the same equation gives θ.
