A chord is any straight segment connecting two points on a circle's edge, and its length is fully determined by the circle's radius plus one more piece of information — either the central angle it subtends or its perpendicular distance from the center. This calculator handles both forward calculations and the reverse: solving for the radius when the chord and angle are known.
How the Chord Length Calculator works
Draw the two radii from the circle's center to each end of the chord. Together with the chord, they form an isosceles triangle (both radii have length r). Splitting that triangle down the middle with a perpendicular bisector creates two right triangles, each with hypotenuse r, one leg equal to half the chord, and the angle θ/2 at the center. That immediately gives chord = 2r·sin(θ/2). The same right triangle also relates the perpendicular distance d from the center to the chord: d = r·cos(θ/2), and by the Pythagorean theorem, half the chord = √(r² − d²), giving the equivalent chord = 2√(r² − d²).
Inputs and what they mean
On the From Angle tab, enter the radius (r) and the central angle (θ) in degrees, from just above 0° up to just below 360°. On the From Distance tab, enter the radius and the perpendicular distance (d) from the center to the chord — d must be between 0 (a diameter) and r (a chord that has shrunk to a single point). On the Solve Radius tab, enter a known chord length and its central angle to back-calculate the radius of the circle it belongs to.
Limits and edge cases
The longest possible chord in any circle is the diameter, which occurs at a central angle of exactly 180° or a distance to center of 0. As the central angle approaches 0° or 360°, or the distance to center approaches the radius, the chord length approaches zero. If a distance to center greater than the radius is entered, no valid chord exists — the calculator flags this rather than returning an invalid number, since the chord's perpendicular distance can never exceed the circle's own radius.