Math · Geometry
Arc length calculator.
Give any two of radius, central angle and arc length. Get the third, plus the chord and the height of the arc.
Inputs
Arc
Enter your values.
How do you find arc length?
An arc is the part of a circle’s edge cut off by a central angle. Its length is s = rθ with θ in radians, or s = 2πr × (angle / 360°) with the angle in degrees. Turn it around to find the angle, θ = s / r, or the radius, r = s / θ. The straight line joining the ends of the arc is the chord, 2r sin(θ/2), and the height of the arc above the chord is the sagitta, r(1 − cos(θ/2)).
Worked example. A 60° arc on a circle of radius 10 is 10.472 long, with a chord of 10 and a sagitta of 1.3397. Going the other way, an arc of 8 on a radius of 5 spans 1.6 rad (91.6732°).
Frequently asked questions
What is the arc length formula?
Arc length s = r × θ when θ is in radians. In degrees, s = (θ / 360) × 2πr.
How do I find the central angle from the arc length?
Divide the arc length by the radius to get the angle in radians (θ = s / r), then multiply by 180/π for degrees. For s = 8 and r = 5 that is 1.6 rad, or 91.6732°.
What is the difference between arc length and chord length?
The arc follows the curve; the chord is the straight line between its ends, so it is always shorter. For a 60° arc on radius 10 the arc is 10.472 and the chord is 10.
Can the angle be more than 180 degrees?
Yes, up to a full 360° turn. Beyond 180° the arc is the major arc and the chord still joins its two ends.
Sources
Checked
Arc length s = R*theta, chord 2R sin(theta/2), sagitta R(1 - cos(theta/2)) and the segment formulas are from the Circular Segment page. Degree to radian conversion is pi/180 (NIST SP 811 Table 6). Runs entirely in your browser. Nothing you type is sent or stored.
Scope
Results are exact mathematics rounded for display, so they are only as good as the measurements you enter. Any length unit works as long as every value uses the same one; areas come out in that unit squared and volumes in that unit cubed. For homework, DIY layout and estimating, not a substitute for a surveyor, engineer or your teacher's required method.