Free Chord Length Calculator

Last updated October 13, 2026 · Report an error

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A chord is a straight line segment connecting any two points on a circle's circumference — the circle's diameter is simply its longest possible chord, passing through the center. Chord length calculations are common in engineering and construction (laying out a curved section of road, railway track, or arch where only the radius and angle are known), machining (calculating flat-to-flat distances on cylindrical stock), and astronomy (measuring apparent separations along circular paths).

How it works

Choose which information you have. If you know the central angle θ that the chord subtends, the chord splits into two right triangles when you draw a line from the center to the chord's midpoint, giving chord=2r·sin(θ/2). If you instead know the perpendicular distance d from the center to the chord, the Pythagorean theorem relates the radius, that distance, and half the chord: chord=2√(r²-d²).

Enter given information.
Enter radius.
Enter central angle (for angle mode).
Enter angle unit.
Enter perpendicular distance (for distance mode).
Click Calculate to see your results.

Examples

r=10, θ=60°

Chord = 2(10)sin(30°) = 20(0.5) = 10.

r=13, perpendicular distance d=5

Chord = 2√(13²-5²) = 2√(169-25) = 2√144 = 2(12) = 24.

Common mistakes to avoid

Entering a perpendicular distance larger than the radius, which isn't geometrically possible (the line wouldn't reach the circle at all).
Using the full central angle instead of half of it inside the sine function — the correct formula uses sin(θ/2), not sin(θ).
Confusing chord length with arc length — they represent different paths between the same two points.

Frequently asked questions

The chord length is 0 — the "chord" collapses to a single point where the line is tangent to the circle rather than crossing through it.

The diameter — a chord passing exactly through the center, with central angle 180° and perpendicular distance 0. Every other chord is shorter.

They measure different things for the same central angle: arc length is the curved distance along the circle's edge (rθ), while chord length is the straight-line shortcut between the same two endpoints — the chord is always shorter than the corresponding arc.

Yes — rearranging the formula gives θ=2·arcsin(chord/(2r)), though this specific calculator is set up to solve for the chord itself rather than working backward.

Drawing a radius to one end of the chord, together with the perpendicular from the center to the chord, forms a right triangle whose hypotenuse is the radius r, one leg is the perpendicular distance d, and the other leg is half the chord — so half the chord equals √(r²-d²), and doubling it gives the full chord length.

Only within the same circle — a chord of a given length subtends a smaller central angle in a larger circle and a larger central angle in a smaller circle, since the same straight-line length represents a different fraction of the circumference depending on the radius.