Circle Chord Length Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The length of a circle's chord is calculated with L = 2r × sin(θ/2), where r is the circle radius and θ the central angle formed by the two radii bounding that chord. For a circle of radius 5 cm and a central angle of 60°, the chord is exactly 5 cm — the same length as the radius.
Explanation
A chord is the straight line segment connecting two points of a circle; it differs from the arc, which follows the curvature of the circle between those same two points rather than connecting them in a straight line — the chord is always shorter than the corresponding arc, except in the limiting case of a zero angle where the two merge into a point. This formula is easily proved by splitting the isosceles triangle formed by the two radii and the chord into two equal right triangles, each with angle θ/2 at the center: half the chord is then r×sin(θ/2), hence the full formula by doubling this result. A notable special case lets this formula be checked without a calculator: at a central angle of exactly 60°, the chord has precisely the same length as the radius, since the two radii and the chord then form a perfect equilateral triangle (all angles are 60°, so all sides are equal) — exactly what this calculator's first test case confirms. This same radius-angle pair is also used to calculate the area of a circular sector bounded by those two radii, a different quantity (an area, not a length) but built from the same two starting values.
Example: radius 10 cm, central angle of 90°
Inputs
Radius: 10 cm. Central angle: 90°.
Calculation
L = 2 × 10 × sin(45°) = 20 × 0.7071 ≈ 14.14 cm.
Result
The length of this chord is about 14.14 cm.
Frequently asked questions
What is the difference between a chord and an arc of a circle?
The chord is the shortest straight segment connecting two points of a circle, while the arc follows the curvature of the circle between those same two points, staying on the circle itself. The arc is always longer than the corresponding chord (except for a zero angle, where both reduce to a point), since a curved line between two points is always longer than the straight segment connecting them directly.
Why does the chord equal exactly the radius at an angle of 60°?
Because at that precise angle, the triangle formed by the two radii and the chord becomes equilateral: the two radii are by definition equal in length (the circle radius), and a 60° angle between them mathematically forces the third side (the chord) to have exactly that same length, since all three angles of an equilateral triangle are 60°.
What is the longest possible chord in a circle?
The longest possible chord is the diameter of the circle, obtained for a central angle of 180°: the formula then gives L = 2r × sin(90°) = 2r, exactly the diameter. Beyond 180°, the central angle describes the chord "from the other side" of the circle, but the length of the chord itself starts decreasing again symmetrically.