How this instrument works
A chord is the straight segment joining two points on a circle — the direct line across, not the curved arc between them. Draw the two radii to those points and the chord itself, and an isosceles triangle appears: two sides of length r meeting at the central angle θ. Drop a perpendicular from the center to the chord and it bisects both the angle and the chord at once, leaving two right triangles with hypotenuse r and angle θ ⁄ 2. The side facing that half-angle is half the chord, so half-chord = r·sin(θ ⁄ 2) — double it and chord = 2r·sin(θ ⁄ 2) falls out with no further assumption.
The same length also follows from the law of cosines applied to that isosceles triangle: chord² = r² + r² − 2r²cos θ = 2r²(1 − cos θ), algebraically the same statement as 2r·sin(θ ⁄ 2) once the half-angle identity sin²(θ ⁄ 2) = (1 − cos θ) ⁄ 2 is substituted in. The relationship is old: Hipparchus and later Ptolemy built entire tables of chord(θ) = 2R sin(θ ⁄ 2), with R fixed at 60 in the sexagesimal style of the Almagest, centuries before the word 'sine' existed, using exactly this construction to do the job trigonometric tables do today.
The formula is not monotonic across a full circle, which surprises people used to functions that only grow. As θ climbs from 0° to 180°, sin(θ ⁄ 2) climbs from 0 to 1 and the chord grows to its maximum, the diameter 2r — but push θ past 180° toward 360° and sin(θ ⁄ 2) heads back down, so the chord shrinks again even though the angle keeps increasing, until the two points coincide once more at θ = 360°. A chord can never exceed the diameter; θ = 180° is the one angle where the straight line and the widest possible cut across the circle agree exactly.
- Enter the circle's size into the Radius field, in any length unit — the chord comes back in that same unit.
- Enter the angle between the two radii into Central angle, then set its unit to degrees, radians, or turns from the toggle beside it.
- Read Chord length: the straight-line distance between the two points where those radii meet the circle, not the curved arc.
- Push Central angle to 180° to check the extreme case — Chord length should equal exactly twice the Radius, the diameter.
- Set Central angle to 0° to confirm Chord length collapses to 0, since both radii then point at the same spot on the circle.
Worked example — a 60° chord on a radius-5 circle
Set Radius to 5 and Central angle to 60°, which the sheet converts to 1.0471975511965976 radians before dividing by two. Half the angle is 30°, sin(30°) = 0.5 exactly, so chord = 2 × 5 × 0.5 = 5 — Chord length reads 4.999999999999999, off from a clean 5 only by the last bit of floating-point rounding inside sin(π ⁄ 6).
That result is not a coincidence of the numbers chosen — 60° is the one central angle where the chord always equals the radius, for any circle. The triangle formed by the two radii and the chord has two sides of length r and an included angle of 60°; since the two base angles must be equal and the three angles sum to 180°, each base angle is also 60°, making the triangle equilateral. An equilateral triangle's three sides match, so the chord — the third side — comes out exactly r, here 5, matching the radius on the nose.
Questions
What is the formula for chord length, and where does it come from?
chord = 2r·sin(θ ⁄ 2), with θ the central angle in radians. It comes from bisecting the isosceles triangle formed by the two radii and the chord: the perpendicular from the center splits it into two right triangles with hypotenuse r and angle θ ⁄ 2, so half the chord is r·sin(θ ⁄ 2) and the full chord is double that.
How is chord length different from arc length for the same angle?
Arc length, s = rθ, measures the curved distance along the circle's edge; chord length measures the straight line cutting across between the same two points. The chord is always the shorter of the two — a straight line beats any curved path between two points — and the gap between them grows as θ increases, vanishing only as θ approaches 0.
What is the longest possible chord in a circle?
The diameter. Chord length grows with θ only up to 180°, where sin(θ ⁄ 2) = sin(90°) = 1 and chord = 2r exactly. No straight line drawn between two points on a circle can exceed that width, so 180° marks a genuine ceiling on the formula, not just a convenient checkpoint.
Why does the formula use sin(θ ⁄ 2) instead of sin(θ)?
Because the angle that actually sits inside the right triangle used to derive the formula is half the central angle, not the whole thing — the perpendicular from the center bisects θ on its way to bisecting the chord. Plugging the full angle into sin(θ) instead of sin(θ ⁄ 2) is the single most common slip when doing this by hand.
Does it matter whether I enter the central angle in degrees or radians?
Not on this sheet — the unit toggle converts degrees, radians, or turns to radians before the sine is taken, so any of the three gives the same chord. By hand it matters enormously: treating 60 as if it were already radians gives a different, wrong number than converting 60° properly to π ⁄ 3 radians first.
Can the chord ever be longer than the arc between the same two points?
No. A straight line between two points is always the shortest path connecting them, and the arc is a curved path between those same points, so arc length is at least the chord length for every angle from 0° up to 360°. The two lengths only meet, in the limit, as θ shrinks toward 0, where the curve straightens out and both approach the same tiny value.