Arc Length Calculator

Which value gets solved for

The arc length formula, s = rθ (with θ in radians), has three variables. Fill in any two of radius, angle, or arc length and the calculator rearranges the formula to solve for the third — it detects which field is empty and picks the matching equation automatically. Leave zero or all three fields filled and it asks you to correct the input, since that's not enough (or too much) information to solve.

Worked example

Take a 10 cm radius and a 90° central angle. First convert the angle to radians: 90° × (π/180) = 1.5708 radians. Then apply s = r × θ: 10 × 1.5708 = 15.71 cm. That's the arc length for a quarter of a circle with a 10 cm radius — a quick sanity check, since a full circle's circumference at that radius is 2π(10) ≈ 62.83 cm, and a quarter of that is 15.71 cm, matching the result.

Working backward: if you instead know the radius (10) and the arc length (15.71), the tool computes θ = s/r = 15.71/10 = 1.5708 radians, then converts to degrees (1.5708 × 180/π = 90°) to confirm the same angle.

Reading the diagram

The circle diagram updates as you enter values so you can see the radius line, the swept angle, and the arc itself in different colors. It's a visual check against the numeric result, not a substitute for it — the math in the "Solution" panel is what to trust for exact figures.

Where this comes up

Arc length calculations show up in machining (cutting curved stock to length), road and rail curve design (superelevation and curve layout use arc segments), and any geometry problem involving a sector or circular path rather than a straight line.

Constraints

The angle field is always in degrees; the radius and arc length must use the same distance unit as each other (the tool doesn't do unit conversion). Entering all three fields, or none, returns a prompt rather than a guess.