Chord Length Calculator
Enter a circle's radius and the central angle the chord subtends, and get the chord length instantly — plus the arc length, sagitta and distance from the centre.
Chord length (c = 2·r·sin(θ/2))
7.071068 units
- Arc length (s = r·θ)
- 7.853982
- Sagitta / height (h = r − r·cos(θ/2))
- 1.464466
- Distance centre→chord (d = r·cos(θ/2))
- 3.535534
The chord joins the two points where the radii at angle θ meet the circle. Its length is 2·r·sin(θ/2). A 180° central angle gives a chord equal to the diameter (2r); a 60° angle gives a chord equal to the radius.
How to use this calculator
Type the radius of your circle in any linear unit you like — centimetres, metres, inches or feet — then enter the central angle in degrees. The central angle is the angle at the centre of the circle between the two radii that reach the ends of the chord. The calculator returns the chord length in the same unit as the radius, alongside the arc length, the sagitta (the height of the arc above the chord) and the perpendicular distance from the centre of the circle down to the chord.
How the calculation works
A chord is a straight line joining two points on a circle. Drop the two radii to the ends of the chord and they enclose the central angle θ. Splitting the resulting isosceles triangle down the middle gives two right triangles, each with hypotenuse r and an angle of θ/2 at the centre, so the half-chord is r·sin(θ/2) and the full chord is c = 2·r·sin(θ/2). The same split gives the perpendicular distance from the centre to the chord (the apothem) as d = r·cos(θ/2), which is why the chord can equally be written c = 2·√(r² − d²). The arc that the chord cuts off has length s = r·θ with θ in radians, and the sagitta — the gap between the chord and the arc — is h = r − r·cos(θ/2). The calculator converts your angle from degrees to radians (θ_rad = θ_deg · π/180) before applying these.
Worked example
Take a radius r = 5 and a central angle θ = 90°. Half the angle is 45°, so the chord is c = 2·5·sin(45°) = 10·0.707107 = 7.071068 units — that is 5√2. Check it against the distance form: the distance from the centre to the chord is d = 5·cos(45°) = 3.535534, and 2·√(5² − 3.535534²) = 2·√12.5 = 7.071068, which agrees. The arc is s = 5·(π/2) = 7.853982 and the sagitta is h = 5·(1 − cos 45°) = 1.464466. As a sanity check on the formula itself, a 60° central angle gives c = 2·r·sin(30°) = r — a 60° chord always equals the radius — and a 180° angle gives c = 2r, the full diameter.
Frequently asked questions
What is the formula for the length of a chord?
When you know the radius r and the central angle θ that the chord subtends, the length is c = 2·r·sin(θ/2), with θ measured in the same angle unit your sine function expects (this calculator takes degrees and converts internally). If instead you know the perpendicular distance d from the centre of the circle to the chord, the length is c = 2·√(r² − d²). Both formulas describe the same chord.
What is a central angle?
The central angle is the angle formed at the centre of the circle by the two radii that run out to the two ends of the chord. It is different from an inscribed angle, which sits on the circle itself — for the same chord, the inscribed angle is exactly half the central angle. This calculator uses the central angle, entered in degrees from 0 to 360.
Why does a 60° chord equal the radius?
Substitute θ = 60° into c = 2·r·sin(θ/2): you get c = 2·r·sin(30°) = 2·r·0.5 = r. Geometrically, the two radii and the chord form an equilateral triangle, so all three sides — including the chord — are equal to r. This is also why a regular hexagon inscribed in a circle has a side length equal to the circle's radius.
What is the sagitta of a chord?
The sagitta (Latin for "arrow") is the height of the arc above the chord — the distance from the midpoint of the chord straight out to the arc. It equals h = r − r·cos(θ/2), or equivalently r minus the distance from the centre to the chord. Builders and machinists use it to measure or lay out curves when only the chord and the arc height are accessible.
How long is a chord for a full or zero angle?
At θ = 180° the two radii point in opposite directions, so the chord passes through the centre and equals the diameter, 2r. At θ = 360° the chord's endpoints meet back at the same point, giving a length of 0, and at θ = 0° there is no separation, also 0. The chord is longest — equal to the diameter — exactly at 180°.
What units does the answer come in?
The chord length, arc length, sagitta and centre-to-chord distance all come back in the same linear unit you used for the radius, because they are all lengths scaled by r. Enter the radius in centimetres and every output is in centimetres; use inches and everything is in inches. The central angle stays in degrees and does not carry a length unit.