About the Chord Length Calculator
This chord length calculator finds the straight-line distance between two points on a circle. Give it the radius together with the central angle, the sagitta (the height of the arc above the chord) or the distance from the centre to the chord, and it returns the chord length along with the arc length, segment area and every other measurement.
It also works backwards from the two measurements you can take with a tape on a real curve: the chord (span) and the sagitta (rise). That gives the radius of the arc — exactly what woodworkers, masons, bridge and road designers, and anyone building an arched window, curved wall or eyebrow dormer needs to set out a curve.
Lengths can be in any unit, and angles are in degrees. A sagitta larger than the radius describes the major segment (more than half the circle), which is handled correctly.
With the default inputs, the chord length is 10. Change any value above to recalculate instantly.
How to use the chord length calculator
- 1Choose the pair of values you know.
- 2Enter the lengths in one unit and the angle in degrees.
- 3Read the chord length, or the radius if you entered a chord and sagitta.
- 4Use the arc length and segment area for material estimates.
Formula and method
Draw lines from the centre to both ends of the chord and you get an isosceles triangle with two sides equal to the radius and apex angle θ. Dropping a perpendicular from the centre bisects the chord, so half the chord is r·sin(θ/2) and the distance from the centre to the chord is d = r·cos(θ/2). The sagitta is what is left of the radius: s = r − d.
Rearranging the right triangle (c/2)² + (r − s)² = r² gives the radius from a chord and sagitta, r = (c²/4 + s²) ÷ 2s — the classic arch formula. The arc length is r·θ with θ in radians, and the circular-segment area between the chord and the arc is ½r²(θ − sin θ).
- c
- Chord length
- r
- Radius
- θ
- Central angle subtended by the chord
- s
- Sagitta — height of the arc above the chord midpoint
- d
- Perpendicular distance from the centre to the chord
Worked examples
Radius 10 with a 60° central angle
c = 2 × 10 × sin 30° = 10 — at 60° the chord equals the radius. The arc is 10 × π/3 ≈ 10.47, and the sagitta is 10 − 10·cos 30° ≈ 1.34.
Arched window: 48 in span, 6 in rise
r = (48²/4 + 6²) ÷ (2 × 6) = (576 + 36) ÷ 12 = 51 in. The arch subtends about 56.1° and its curved edge is about 49.98 in long.
Radius 5 with a sagitta of 2
The chord is 5 − 2 = 3 from the centre, so c = 2 × √(25 − 9) = 8 — a 3-4-5 triangle on each side.
Radius 13, chord 5 units from the centre
Half the chord is √(13² − 5²) = 12, so the chord is 24 and the sagitta is 13 − 5 = 8.
Frequently asked questions
What is the chord length formula?+
c = 2r·sin(θ/2), where r is the radius and θ the central angle. If you know the distance d from the centre instead, use c = 2√(r² − d²).
What is a sagitta?+
The sagitta is the height of an arc: the distance from the midpoint of the chord to the arc, measured at right angles to the chord. It is also called the rise or arc height.
How do I find the radius of an arc from its width and height?+
Use r = (c²/4 + s²) ÷ (2s), with c the chord (width) and s the sagitta (height). A 48-inch arch that rises 6 inches has a radius of 51 inches.
What is the longest chord in a circle?+
The diameter. It passes through the centre, so d = 0 and the chord length is 2r, with a central angle of 180°.
What is the difference between a chord and an arc?+
A chord is the straight line between two points on the circle; the arc is the curved path along the circle between the same points. The arc is always at least as long as the chord.