Chord Length Calculator
Calculate the chord length of a circle from the radius plus the central angle, distance from the centre, arc height or arc length, with segment area.
A chord is the straight line joining two points on a circle. Its length depends on the radius and on how far the chord sits from the centre, which you can describe by the central angle, the perpendicular distance from the centre, the height of the arc above the chord, or the arc length. Enter the radius and any one of these, and the calculator gives the chord together with the other three and the area of the circular segment.
How to use this tool
- Choose which measurement you have besides the radius.
- Enter the radius and that measurement; any length unit works.
- For an angle, pick degrees or radians.
- Read the chord length and the related values.
Formula
c = 2r · sin(θ/2) = 2√(r² − d²) = 2√(2rh − h²)
r is the radius, θ the central angle that the chord subtends, d the perpendicular distance from the centre to the chord, and h the arc height (sagitta), with h = r − d for a minor segment and h = r + d for a major segment. From an arc length s, θ = s ÷ r. The segment area between the chord and the arc is r²(θ − sin θ) ÷ 2. The longest possible chord is the diameter, 2r, at θ = 180°.
Example
A circle of radius 10 with a central angle of 60°: c = 2 × 10 × sin 30° = 10. The chord is 10 × cos 30° = 8.660254 from the centre, the arc height is 10 − 8.660254 = 1.339746, the arc length is 10.471976, and the segment area is 100 × (1.0471976 − 0.8660254) ÷ 2 = 9.058607.
Frequently asked questions
How do I find a chord from the radius and the distance to the centre?
Half the chord, the distance d and the radius form a right triangle, so c = 2√(r² − d²). For r = 5 and d = 3, c = 2 × 4 = 8.
Where is chord length used in practice?
Chord and sagitta are how builders, machinists and road designers measure curves whose centre they cannot reach: the span of an arch, the width of a cut across a pipe, or the offset of a curved kerb.
Can the arc height be more than the radius?
Yes, up to the full diameter. The chord then cuts off more than half the circle, the central angle is over 180°, and c = 2√(2rh − h²) still applies.
Which is longer, the chord or the arc?
The arc, always. For small angles they are nearly equal: at 10° on a radius of 10 the chord is 1.743115 and the arc is 1.745329.
Last updated: 2026-10-30