Circle Arc Length Calculator

Arc length equals radius times theta
rθs

Solution

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How It Works

An arc is a portion of a circle's circumference. The arc length formula s = rθ is one of the simplest in geometry: multiply the radius by the central angle in radians. This calculator handles degree-to-radian conversion automatically.

You can solve for the arc length, the radius, or the central angle. Enter any two known values and click Calculate to find the third.

Example Problem

A circle has radius 10 and a central angle of 90°. Find the arc length.

  1. Convert 90° to radians: 90 × π/180 = π/2 ≈ 1.5708 rad
  2. s = 10 × 1.5708 = 15.708

The arc is one-quarter of the full circumference (2π×10 ≈ 62.83), so 62.83/4 ≈ 15.71 ✓

Frequently Asked Questions

What is arc length?

Arc length is the distance along the curved portion of a circle between two points on the circumference. It depends on the radius and the central angle subtended by the arc.

How is arc length related to circumference?

The full circumference is the arc length for a complete 360° (2π radian) rotation: C = 2πr. Any arc is a fraction of this, proportional to the central angle: s = (θ/2π)×C = rθ.

Why must the angle be in radians for s = r theta?

A radian is defined as the angle where the arc length equals the radius (s = r when θ = 1 rad). This definition makes the formula s = rθ exact. For degrees, you must first convert: θ_rad = θ_deg × π/180.

What is the arc length of a semicircle?

A semicircle has a central angle of π radians (180°). Its arc length is s = rπ, which is half the circumference. For r = 10, the semicircular arc is 10π ≈ 31.42.

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