About the Arc Length Calculator
This arc length calculator measures the curved distance along part of a circle. Enter the radius and the central angle to get the arc length, or switch modes to find the central angle from a radius and arc length, or the radius from an arc length and angle. Each answer also shows the sector area and the straight-line chord between the arc’s ends.
Students use it for circle geometry and radian-measure problems. It is also practical for bending pipe or trim to a curve, laying out a curved path or garden edge, working out the distance travelled around a track, or setting out a curved wall where the radius and span are known.
Angles can be entered in degrees or radians. The arc length is in the same unit as the radius. For angles above 360° the arc wraps around the circle more than once, so the calculator limits the sector and chord to a single turn.
With the default inputs, the answer is 10.472. Change any value above to recalculate instantly.
How to use the arc length calculator
- 1Choose what you want to solve for.
- 2Pick degrees or radians for the angle.
- 3Enter the two known values.
- 4Read the answer, plus sector area and chord length.
Formula and method
A radian is defined so that an angle of one radian cuts off an arc exactly one radius long, which makes the arc length simply s = rθ when θ is in radians. In degrees the same idea reads s = 2πr × θ/360: the arc is the same fraction of the circumference as the angle is of a full turn.
Rearranging gives θ = s ÷ r and r = s ÷ θ for the other two modes. The sector area, the pie-slice between the arc and the centre, is ½r²θ, and the chord — the straight line joining the ends of the arc — is 2r sin(θ/2), which is always a little shorter than the arc.
- s
- Arc length
- r
- Radius of the circle
- θ
- Central angle (radians in s = rθ)
Worked examples
Radius 10, central angle 60°
60° is π/3 radians, so s = 10 × π/3 ≈ 10.47. The sector area is ½ × 100 × π/3 ≈ 52.36, and the chord equals the radius because the triangle formed is equilateral.
Find the angle: radius 8, arc 12
θ = s ÷ r = 12 ÷ 8 = 1.5 radians, which is about 85.94°. The sector area is ½ × 64 × 1.5 = 48.
Find the radius: arc 20 over 2 radians
r = s ÷ θ = 20 ÷ 2 = 10. The chord is 2 × 10 × sin 1 ≈ 16.83, noticeably shorter than the 20-unit arc.
Frequently asked questions
What is the formula for arc length?+
With the angle in radians, arc length s = r × θ. With the angle in degrees, s = 2πr × (θ ÷ 360). Both give the same answer.
How do I find arc length without the angle?+
You need a second measurement. If you know the chord length c and radius r, the angle is θ = 2·arcsin(c ÷ 2r); then s = rθ. Alternatively use the sector area: θ = 2A ÷ r².
What is the difference between arc length and chord length?+
Arc length follows the curve of the circle; chord length is the straight line between the arc’s two endpoints. The chord is always shorter, and the gap grows with the angle.
Why are radians used in the arc length formula?+
A radian is defined as the angle that subtends an arc equal to the radius, so arc length is just radius × angle. Using degrees requires the extra 2π/360 conversion factor.