Circle Length Calculator — Arc Length from Radius & Angle
Enter radius and central angle (degrees or radians) to find the arc length — the curved distance along the circle — plus sector area and chord length.
Angle unit
Curved distance along the circle edge subtended by the central angle
- 1
Convert angle to radians
60 × π ÷ 180 = 1.047198 - 2
Arc length = r × θ
10 × 1.047198 = 10.4720
How does this calculator work?
Arc length = r × θ with θ in radians. In degrees: arc = πrθ°/180. A 60° arc on a circle of radius 10 has length 10×(π/3) ≈ 10.47. Enter radius and angle to also get sector area (½r²θ) and chord length (2r·sin(θ/2)).
Formula
How this is calculated
The arc length is the portion of a circle's circumference subtended by a given central angle θ. Since the full circumference 2πr corresponds to a full rotation of 2π radians, any arc is the fraction θ/(2π) of the circumference — which simplifies to rθ when θ is expressed in radians. This is the cleanest form of the formula and the standard used in calculus and physics.
If you enter degrees, the calculator converts them to radians first (multiply by π/180) before applying the rθ formula. The same angle also defines a sector (pie-slice region) with area ½r²θ and a chord (the straight line connecting the two arc endpoints) with length 2r·sin(θ/2), derived from the half-angle in the isosceles triangle formed with the two radii.
Arc length is always greater than or equal to chord length for the same angle; they converge as θ → 0. At θ = 2π (360°) the arc equals the full circumference. These relationships make arc length central to problems in engineering (gear tooth profiles, belt drives) and physics (circular motion, pendulum displacement).
Frequently asked questions
Arc length is the curved distance along the circle between two points. Chord length is the straight-line distance between the same two points. Arc length ≥ chord length, with equality only at θ = 0. For small angles they are nearly equal; for θ = 180° the chord is 2r but the arc is πr ≈ 3.14r.
Arc length is a fraction of circumference: arc = (θ/2π) × 2πr = rθ. A quarter circle (90°, π/2 rad) has arc = πr/2, exactly one quarter of the circumference 2πr.
Yes. If the object completes ω radians per second, a point at radius r travels at tangential speed v = rω — which is the arc length per unit time. The rθ formula is the displacement version: d = rθ for rotation by θ radians.
Also known as
TG we-Calculate Editorial Team. (2026). Circle Length Calculator — Arc Length from Radius & Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circle-length-calculator
TG we-Calculate Editorial Team. "Circle Length Calculator — Arc Length from Radius & Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/circle-length-calculator.
TG we-Calculate Editorial Team, "Circle Length Calculator — Arc Length from Radius & Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circle-length-calculator
@misc{wecalculate_circle_length_calculator, title = {Circle Length Calculator — Arc Length from Radius & Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circle-length-calculator}}, year = {2026}, note = {TG we-Calculate} }
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