Chord Length Calculator - Find Chord, Radius, or Central Angle

A chord is one of the fundamental line segments in circle geometry. It connects any two distinct points on a circle's circumference and divides the interior of the circle into two regions. The diameter is the special case of a chord that passes through the center, and it is the longest chord in any circle. Every other chord is strictly shorter than the diameter and subtends a central angle strictly less than 180°. The relationship between chord length, radius, and central angle is captured by the formula c = 2r × sin(θ/2), where c is the chord length, r is the radius, and θ is the central angle in radians. This formula derives from the law of cosines. If you draw radii to both endpoints of the chord, you create an isosceles triangle with two sides of length r and the included angle θ. Applying the law of cosines gives c² = 2r²(1 − cos θ). Using the half-angle identity 1 − cos θ = 2 sin²(θ/2), this simplifies to c = 2r sin(θ/2). The calculator supports three modes corresponding to the three ways you can rearrange this formula. In 'Find Chord Length' mode, you supply r and θ and the calculator evaluates 2r sin(θ/2) directly. In 'Find Radius' mode, you supply c and θ and the calculator evaluates r = c / (2 sin(θ/2)), which is useful when you have physically measured a chord and know the arc spans a specific angle. In 'Find Central Angle' mode, you supply r and c and the calculator evaluates θ = 2 arcsin(c / (2r)), which yields the angle whose sine is c/(2r). Angle units can be freely switched between degrees and radians. Degrees are more intuitive for most users and are the default, but engineering and scientific work often uses radians because arc length = r × θ only holds in radians. The calculator converts internally so you can use whichever system your source data uses. Practical applications of chord calculations are numerous. In structural engineering, arch bridges are designed using the relationship between span (chord), rise, and radius of curvature. In mechanical engineering, involute gear tooth geometry depends on chord calculations across pitch circles. Surveyors use chord length combined with arc measurements to compute distances when curves interrupt line-of-sight measurements. In woodworking, a craftsperson who needs to cut a curved piece of wood to span a certain distance at a certain angle uses exactly this formula. Navigation historically used chord tables — precursors to modern trigonometric tables — to compute distances on spherical surfaces. This calculator handles any valid input combination within the physical constraints: radii must be positive, chord lengths must be positive and at most twice the radius, and angles must be positive and less than 360° (or 2π radians). Results are rounded to eight decimal places, which is more than sufficient precision for any practical application.