Chord Length Calculator

Background.

The Quanta chord length calculator finds the straight-line distance across a circular arc, and it takes four different routes in because real problems arrive in four different shapes. Give it a radius and a central angle, a radius and the distance from the centre to the chord, a radius and the height the arc rises above the chord, or — the useful one — a span and a rise, and let it hand you back the radius. Every route returns the same eight numbers: chord, radius, apothem, sagitta, central angle in degrees and radians, arc length, and the area of the circular segment the chord cuts off.

With the defaults — a radius of 10 and a central angle of 60° — the chord comes back as exactly 10. That is not a rounding coincidence. Euclid proved it in Book IV, Proposition 15, whose corollary states plainly that the side of an inscribed regular hexagon equals the radius of the circle, so six 60° chords laid end to end close the circle. The apothem is 8.6602540378, the sagitta is 1.3397459622, the arc measures 10.471975512 against the chord's 10, and the segment it cuts off has an area of 9.0586073706.

The fourth mode is the reason this page is more than a lookup table. Arches, culverts, road curves, boat frames and rolled sheet metal are all specified by span and rise, never by radius and angle — nobody measures a central angle on a building site. Enter a 12-unit span and a 2-unit rise and the calculator returns a radius of exactly 10, which is the number you set a trammel or a beam compass to. It works for shallow arcs and for arcs past a half circle equally: a 12-unit span with a 16-unit rise gives a radius of 9.125 and a central angle of 277.7758191217°, which is a major segment holding more than half the circle.

Three words are worth pinning down before you read the answer, because they get swapped constantly. The chord is the straight line between the arc's two ends. The sagitta — Latin for arrow, with the chord as the bowstring — is how far the arc bulges above that line at its midpoint. The apothem is the perpendicular distance from the circle's centre down to the chord, and for a minor arc the radius is simply the apothem plus the sagitta. All three are reported here, so whichever one your drawing labels, you can read it off directly.

One convention needs stating up front rather than in a footnote. Given only a radius and an apothem there are two possible arcs — the small one and the large one on the other side of the same chord — and this calculator reports the small one, which is the universal convention. If you want the major arc, use the sagitta instead: that mode covers the whole range from a barely curved sliver up to very nearly the entire circle, and the segment area grows accordingly. The apothem is always reported as an unsigned distance, so for a major segment it tells you how far the centre is from the chord without implying which side it sits on.

What is chord length calculator?

A chord is the straight line segment joining two points on a circle. Wolfram MathWorld puts it generally: "In plane geometry, a chord is the line segment joining two points on a curve", most often used of a segment whose ends lie on a circle. The longest chord of any circle is its diameter, which is the chord that passes through the centre.

Every chord comes with a family of related measurements, and MathWorld's Circular Segment entry gives the formulas that connect them. With R the radius and θ the central angle the chord subtends, the chord length is Equation (6), a = 2R·sin(θ ÷ 2). The perpendicular distance from the centre to the chord — the apothem — is Equation (3), r = R·cos(θ ÷ 2), and Equation (8) gives the same length the other way round, a = 2√(R² − r²). The sagitta, the height of the arc above the chord, follows from Equation (1), R = h + r, so h = R(1 − cos(θ ÷ 2)). The area of the circular segment the chord cuts off is Equation (15), A = ½R²(θ − sin θ), with θ in radians.

The relation that needs no trigonometry at all is older by two thousand years. Euclid's Elements, Book III, Proposition 35 — the intersecting chords theorem — states that when two chords cross, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other. Apply it to a chord and the diameter through its midpoint and the two pieces of the diameter are h and 2R − h while both halves of the chord are c ÷ 2, so (c ÷ 2)² = h(2R − h). Rearranged, that is R = (h² + (c ÷ 2)²) ÷ 2h, the formula this page's fourth mode uses to recover a radius from a span and a rise. It is also the identity the test suite uses to check every answer this calculator produces, in all four modes, since it shares no arithmetic with the sine and cosine route.

One further connection is worth naming. An isosceles triangle whose two equal sides are radii and whose included angle is θ has the chord as its third side, so the law of cosines gives c² = 2R²(1 − cos θ) directly. It is the same formula as 2R·sin(θ ÷ 2) after a half-angle identity, and it is why a chord calculation and a triangle calculation are secretly the same computation approached from opposite ends.

How to use this calculator.

1. 01Pick the mode that matches what you have measured. Radius and central angle is the textbook case; span and rise is the one that matches a drawing or a site measurement.
2. 02In the first mode, enter the radius and the central angle in degrees. The angle must be greater than zero and less than a full turn — at either end the chord's two endpoints meet and there is nothing to measure. If you are working in radians, multiply by 180 ÷ π first; the answer comes back in both units either way.
3. 03In the second mode, enter the radius and the perpendicular distance from the centre to the chord. Zero is allowed and means the chord passes through the centre, making it a diameter.
4. 04In the third mode, enter the radius and the arc's rise. This is the mode to use if your arc is more than a half circle, because the rise can go all the way up to the diameter.
5. 05In the fourth mode, enter the span between the arc's two ends and the rise at its midpoint. The radius comes back as an answer — set your trammel or beam compass to it.
6. 06Read the chord length at the top, then the radius, apothem and sagitta beneath it. Whichever of those three your drawing labels, it is on the page.
7. 07Compare the arc length with the chord if you are deciding whether a curve can be cut straight. For the default 60° arc the difference is 0.471975512 in 10, about 4.7 per cent.
8. 08Use the segment area for a fill calculation — it is exactly the cross-section of liquid in a horizontal cylindrical tank filled to a given depth, with the depth entered as the sagitta.

The formula.

Drop a perpendicular from the centre to the chord. It bisects the chord and splits the central angle in two, leaving a right triangle whose hypotenuse is the radius, whose short side is half the chord, and whose angle at the centre is θ ÷ 2. From that single triangle everything follows: half the chord is R·sin(θ ÷ 2) and the perpendicular itself is R·cos(θ ÷ 2). Double the first and you have the chord; the second is the apothem, and the sagitta is whatever is left of the radius, R − R·cos(θ ÷ 2).

Each mode reduces to the central angle first and then rebuilds everything from it. The first mode has the angle already. The second inverts the apothem relation, θ = 2·arccos(r ÷ R), which lands between 0° and 180° and is therefore the minor arc. The third inverts the sagitta relation, θ = 2·arccos((R − h) ÷ R), which covers the full range up to a whole turn. The fourth first recovers the radius as R = (h² + (c ÷ 2)²) ÷ 2h and then proceeds as the third. That radius formula never fails: writing q for half the chord, 2R = (h² + q²) ÷ h = h + q² ÷ h, which always exceeds h, so any positive span with any positive rise describes exactly one circle.

Working the default all the way through: R = 10 and θ = 60°, so θ ÷ 2 = 30°. The chord is 2 × 10 × sin 30° = 2 × 10 × 0.5 = 10 exactly. The apothem is 10 × cos 30° = 10 × 0.866025403784438646 = 8.660254037844386, displayed as 8.6602540378. The sagitta is 10 − 8.660254037844386 = 1.339745962155614, displayed as 1.3397459622. In radians the angle is π ÷ 3 = 1.047197551196598, so the arc is 10 × 1.047197551196598 = 10.471975511965977, displayed as 10.471975512. The segment area is ½ × 100 × (1.047197551196598 − sin 60°) = 50 × (1.047197551196598 − 0.866025403784439) = 50 × 0.181172147412159 = 9.058607370607955, displayed as 9.0586073706.

Every one of those numbers can be checked without a sine table, using a theorem from about 300 BCE. Euclid's Elements, Book III, Proposition 35 gives (c ÷ 2)² = h(2R − h) for any chord. Here that is 5² = 25 on the left, and 1.339745962155614 × (20 − 1.339745962155614) = 1.339745962155614 × 18.660254037844386 = 25.000000000000000 on the right. Book IV, Proposition 15 and its corollary — the side of an inscribed regular hexagon equals the radius — fixes the headline answer independently: a 60° chord in a radius-10 circle must be exactly 10, and it is. A third check comes from the law of cosines on the isosceles triangle with two sides of 10 and an included angle of 60°: c² = 200(1 − cos 60°) = 200 × 0.5 = 100, so c = 10 again.

Rounding stage: every intermediate is carried at 40 significant digits and rounded once, at the return boundary, to 10 decimal places, with π computed as arccos(−1) at that precision rather than typed as a literal. The central angle is the pivot and is never rounded before the chord, apothem, sagitta, arc and segment area are derived from it — rounding it first would destroy the exact 10, because sin(30.0000000001°) is not 0.5, and would leave Euclid's III.35 check failing in the ninth decimal place in span-and-rise mode where the radius is itself derived. Nothing on this page is sorted into bands or graded, so no wording depends on a rounded value.

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 its tests check every answer 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, and is asserted as such in chord-length.test.ts.
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.