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Measure how far you travel along a circle: arc length from radius and central angle, or solve for radius or angle when you know the curve. Works in degrees and radians with the same formulas your textbook uses—plus clear steps and real-world examples below.
Last updated: March 21, 2026
Three solve modes, two angle units, and algebra that matches standard geometry and calculus courses—so you can verify homework or estimate real curved paths in consistent units.
Arc length is the distance along the curved part of a circle between two points on the circumference—not the straight chord through the interior. If you unwrapped that curve into a straight segment, its length is s.
It matters anywhere a path follows a circular or nearly circular track: road curves, pulleys and belts, gear teeth spacing along pitch circles, satellite ground tracks, and calculus problems involving parametrized curves (where arc length generalizes beyond circles).
Do not confuse arc length with arc measure: the latter is only the angle at the center (for example 90°). Two circles can share the same central angle but have very different arc lengths because the larger circle has a longer circumference for the same "slice" of the pie.
Use one consistent unit system for radius and arc length (meters with meters, inches with inches). The angle must match the formula: radians with s = rθ, or degrees with the 360° full-turn version below.
The full circumference is 2πr. Multiply by the fraction of a full turn θ/360 to get the length of that portion. Equivalent compact form: s = πrθ / 180.
Inverse formulas (same unit choices): from s = rθ you get r = s/θ and θ = s/r in radians. In degrees, solving for radius gives r = (180 × s) / (π × θ) and for angle θ = (180 × s) / (π × r)—exactly what the calculator uses when you switch modes.
Numbers below use the same formulas as the calculator; units are arbitrary as long as radius and arc length match.
Same radius, wider angle → longer arc. Same angle, larger radius → longer arc. Tables use r = 10 (angle sweep) and θ = 45° (radius sweep); values are arc length in the same unit as radius.
| Central angle θ | Arc length s | % of circumference |
|---|---|---|
| 30° | 5.236 | 8.3% |
| 60° | 10.472 | 16.7% |
| 90° | 15.708 | 25.0% |
| 180° | 31.416 | 50.0% |
| Radius r | Arc length s | vs r = 5 |
|---|---|---|
| 5 | 3.927 | 1.00× |
| 10 | 7.854 | 2.00× |
| 15 | 11.781 | 3.00× |
In the calculator, choose arc length mode, set radius 5, angle 72°, and degrees. You should get 6.2832 — because 72/360 = 0.2, one-fifth of the circumference 2πr.
Share this step-by-step calculator with your study group or classmates to instantly verify homework answers.
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