Chord Length Calculator

Background.

A bricklayer turning a segmental arch over a 3.6 m opening, with the crown 0.6 m above the springing line, needs one number the drawing does not give: the radius to set the trammel to. It is 3 m, and finding it is a chord problem. So is cutting a curved steel flat to length, laying out a curved path, or working out how much liquid sits in a horizontal tank at a given depth.

A chord is the straight line between two points on a circle. Three other lengths hang off it. The radius runs from the centre to the circle. The apothem is the perpendicular distance from the centre to the chord. The sagitta, Latin for arrow with the chord as the bowstring, is how far the arc rises above the chord at its midpoint. Along the line through the centre and the chord's midpoint, the radius is the apothem plus the sagitta whenever the arc is less than a half circle. Add the central angle the chord subtends, and any two of these measurements settle all the rest.

You can start from radius and central angle, radius and apothem, radius and sagitta, or span and rise, the last returning the radius itself. Whichever you choose, you get the chord, radius, apothem, sagitta, the central angle in degrees and radians, the length of the arc, and the area of the circular segment between chord and arc. Lengths come back in whatever unit you entered, and areas in that unit squared.

What is chord length calculator?

In MathWorld's definition, a chord is the line segment joining two points on a curve; on a circle, the longest possible chord is the diameter, which passes through the centre. A chord splits the circle into two arcs, a minor one under 180° and a major one over it, and splits the disc into two circular segments. A segment is the region between chord and arc, which is different from a sector, the pie slice running back to the centre. Every chord calculation rests on one right triangle: the radius as hypotenuse, half the chord as one side and the apothem as the other.

How to use this calculator.

1. 01Choose the mode that matches what you measured. Span and rise is the one for drawings and site work, where nobody measures a central angle.
2. 02For radius and angle, give the angle in degrees, above 0 and below 360. To convert radians, multiply by 180 ÷ π; the result shows both units anyway.
3. 03For radius and apothem, the apothem must be smaller than the radius; zero means the chord passes through the centre and is a diameter. This mode always gives the smaller of the two arcs.
4. 04For radius and sagitta, the rise can go up to, but not reach, the diameter. A rise greater than the radius describes an arc of more than half a circle.
5. 05Use the arc length when cutting material that will be bent to the curve, and the segment area for glazing, fill or tank contents.

The formula.

Drop a perpendicular from the centre to the chord. It cuts the chord in half and splits the central angle θ in two, leaving a right triangle with the radius R as hypotenuse. Half the chord is R·sin(θ⁄2) and the perpendicular is R·cos(θ⁄2), so the chord is c = 2R·sin(θ⁄2), the apothem is r = R·cos(θ⁄2), reported as a positive distance, and the sagitta is h = R − R·cos(θ⁄2). With θ in radians, the arc length is Rθ and the segment area is ½R²(θ − sin θ).

Each mode first finds θ. Radius and apothem give θ = 2·arccos(r ÷ R), which always lands between 0° and 180°, the minor arc. Radius and sagitta give θ = 2·arccos((R − h) ÷ R), which covers everything up to a full turn. Span and rise go through Euclid's intersecting-chords theorem (Elements III.35): the chord and the diameter through its midpoint cut each other so that (c⁄2)² = h(2R − h). Solved for the radius, R = (h² + (c⁄2)²) ÷ 2h. Writing q for half the span, 2R = h + q² ÷ h, which always exceeds the rise, so any positive span and rise define exactly one circle.

One symmetry is worth knowing: a chord is the same for θ and 360° − θ. At a radius of 10, both 60° and 300° give a chord of 10; what changes is the sagitta, 1.3397 against 18.6603, and which segment is reported.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Circular Segment’ (live revision retrieved 2026-07-29). Source for every formula on this page: Equation (1) R = h + r, Equation (3) r = R cos(½θ), Equation (5) r = ½√(4R² − a²), Equation (6) a = 2R sin(½θ), Equation (8) a = 2√(R² − r²), Equation (13) θ = 2 sin⁻¹(a/(2R)), Equation (15) A = ½R²(θ − sin θ) and Equation (17) A = R² cos⁻¹(r/R) − r√(R² − r²). Open access.
2. [2]Euclid, Elements, Book III, Proposition 35 — ‘If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other.’ David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29. Applied to a chord and the diameter through its midpoint this gives (c/2)² = h(2R − h), which is both the source of this page's span-and-rise radius formula and the trigonometry-free identity every answer is checked against.
3. [3]Euclid, Elements, Book IV, Proposition 15 — ‘To inscribe an equilateral and equiangular hexagon in a given circle’, with the corollary ‘the side of the hexagon equals the radius of the circle.’ David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29. This fixes the worked example's chord at exactly 10 in a radius-10 circle, independently of any trigonometric formula.
4. [4]Wolfram Research, MathWorld — ‘Chord’ (live revision retrieved 2026-07-29). Cited for terminology only: the definition ‘In plane geometry, a chord is the line segment joining two points on a curve’, and the naming of R for the radius, a for the chord, r for the apothem and h for the sagitta. Recorded honestly — this entry carries no chord-length formula, so it is not the source of any computation on this page.