How to use the Arc Length Calculator
- Enter the circle's radius.
- Enter the central angle in degrees.
- Read the arc length, measured along the curve.
- Read the chord length, the straight line between the arc's endpoints.
- Use the radian value when feeding the angle into other formulas.
How the calculation works
Arc length is beautifully simple in radians: it is just radius times angle. That is effectively the definition of the radian — the angle for which the arc equals the radius. Working in degrees requires the conversion factor π/180 first, which is why every arc formula quoted in degrees carries that awkward constant while the radian version carries none.
The chord is a different measurement entirely: the straight-line shortcut between the two ends of the arc, given by twice the radius times the sine of half the angle. The chord is always shorter than the arc, and the gap widens as the angle grows. For small angles they are nearly identical — the small-angle approximation behind countless engineering shortcuts — but at 180° the arc is πr while the chord is only 2r, a 57% difference.
Both measurements matter in fabrication. Rolling a steel section into a curve requires the arc length to cut the stock, but setting out the ends on a workshop floor requires the chord. Road and rail alignment design uses chord offsets to lay out curves with a tape, and highway curve tables are built entirely from these two relationships.
arc s = rθ (θ in radians); chord = 2r·sin(θ/2); θ_rad = θ_deg × π/180Source: Radian definition and circular arc relations; see NIST DLMF §4.14 and standard highway curve geometry.
Worked example
A curved handrail follows a 10 m radius through 72°.
- Convert: 72° × π/180 = 1.2566 radians.
- Arc length: 10 × 1.2566 = 12.57 m of rail to bend.
- Chord: 2 × 10 × sin(36°) = 20 × 0.5878 = 11.76 m.
- Difference: 0.81 m more material than the straight-line span suggests.
12.57 m of rail stock required, spanning a 11.76 m chord — order to the arc length, set out to the chord.
Frequently asked questions
How do I find the radius from an arc and a chord?+
There is no closed form; it requires iterating on the relationship s = rθ with chord = 2r·sin(θ/2). Numerical solution is the practical route.
What is the arc length of a full circle?+
The full circumference, 2πr, corresponding to a 360° central angle.
Why is my chord nearly equal to the arc?+
Small angles make sine almost linear, so arc and chord converge. Below about 10° the difference is under 0.13%.
Does this apply to non-circular curves?+
No. Parabolic and spiral transition curves have their own arc-length integrals.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.