Circle Arc Length Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/8/2026
A circle arc's length is calculated with (angle ÷ 180) × π × radius. A 90° arc on a circle with a 5 cm radius therefore measures about 7.85 cm.
Explanation
The length of a circle arc is the portion of the circumference bounded by a given angle, measured along the curve rather than in a straight line. As with the area of a circular sector, this length is simply a fraction of the circle's full circumference (2π × radius), the fraction being the ratio between the arc's angle and the full angle (360°). A useful consistency check: with an angle of 360°, this formula gives back exactly the full circumference of the circle, since the arc then goes all the way around. This concept is used in practice to calculate the length of a curved rail, a section of a running track on a bend, or the distance traveled by the tip of a clock hand or a rotating arm over a given angle — in all these cases, it's a distance measured along a curved path, not the straight-line distance between the two ends of the arc (which would be a chord, always shorter than the arc itself).
Example: a 90° arc, 5 cm radius
Inputs
Radius: 5 cm. Arc angle: 90°.
Calculation
Length = (90 ÷ 180) × π × 5 = 0.5 × π × 5 = 0.5 × 15.708 ≈ 7.85 cm.
Result
This 90° arc on a circle with a 5 cm radius measures about 7.85 cm.
Frequently asked questions
What's the difference between arc length and chord length?
The arc length follows the circle's curve between the two points, while the chord is the straight line directly connecting these same two points. The chord is always shorter than the arc (except at a zero angle, where the two coincide), with the gap between them growing with the angle: at 360°, the arc goes all the way around the circle while the chord... no longer even makes sense, the two points having merged.
How do you calculate the angle if you already know the arc length?
By rearranging the formula: angle in degrees = (arc length ÷ (π × radius)) × 180. This calculator goes from angle to length, but the same principle applies in reverse to find the angle from a measured arc length.
Why divide by 180 and not by 360 like for the sector area?
Because a circle's full circumference equals 2π × radius (not π × radius as for the area), so the fraction corresponding to a given angle is written (angle ÷ 360) × 2π × radius, which simplifies to (angle ÷ 180) × π × radius by canceling the 2 in the numerator with the 360 in the denominator.