Arc length is the distance along part of a circle’s circumference. If you walk along the curved edge from one endpoint to another, that curved distance is the arc length.

Why must the angle be in radians for s = rθ?

The formula s = rθ is built from the definition of radian measure. Radians already express the ratio between arc length and radius, so multiplying the radian angle by the radius gives the arc length directly.

How do I calculate arc length when the angle is in degrees?

First convert degrees to radians by multiplying by π/180. Then use s = rθ. This calculator does that conversion automatically, but showing the radian value helps students follow the step.

Is arc length the same as chord length?

No. Arc length is curved distance along the circle. Chord length is the straight-line distance between the same endpoints. For the same endpoints, the arc is usually longer than the chord unless the angle is extremely small.

Education

Arc Length Calculator

Calculate the arc length of a circle from radius and central angle, with degree/radian conversion, chord length, and step explanation. This page is written for students who need both a fast answer and a clear explanation of the method, so it includes the formula, steps, examples, common mistakes, and practical notes.

Arc length connects angle measure to distance around a circle

Arc length is the distance along the curved edge of a circle. The clean formula is s = rθ, but θ must be in radians. The OpenStax angles lesson explains radian measure through the relationship between an arc, a radius, and the central angle, which is exactly why the formula works.

Formula and example

s = rθ If θ is in degrees: θ radians = degrees × π ÷ 180 Example: r = 6, θ = 75° θ = 75π/180 = 1.309 rad s = 6 × 1.309 = 7.854

Arc length is not the same as chord length. The arc follows the circle, while the chord is the straight line between the endpoints.

Common questions

Arc length is the distance along part of a circle’s circumference. If you walk along the curved edge from one endpoint to another, that curved distance is the arc length.
The formula s = rθ is built from the definition of radian measure. Radians already express the ratio between arc length and radius, so multiplying the radian angle by the radius gives the arc length directly.
First convert degrees to radians by multiplying by π/180. Then use s = rθ. This calculator does that conversion automatically, but showing the radian value helps students follow the step.
No. Arc length is curved distance along the circle. Chord length is the straight-line distance between the same endpoints. For the same endpoints, the arc is usually longer than the chord unless the angle is extremely small.