Chord Calculator — Length, Sagitta, Arc and Segment Area

Formula

How this is calculated

A chord is the straight line connecting two points on a circle's circumference. Its length depends on the radius r and the central angle θ (the angle at the circle's centre between the two radii that reach the chord's endpoints). Because the two radii and the chord form an isosceles triangle, halving the construction gives a right triangle with hypotenuse r and half-angle θ/2, so the half-chord equals r × sin(θ/2) and the full chord is c = 2r × sin(θ/2).

The sagitta h is the perpendicular distance from the midpoint of the chord to the arc — the "bulge" of the arc above the chord. Since the foot of the sagitta lies at distance r × cos(θ/2) from the centre, the sagitta is h = r − r × cos(θ/2) = r × (1 − cos(θ/2)). Sagittae appear in bridge and lens design to describe how much a curved surface departs from flat. Arc length is simply the fraction of the full circumference: L = r × θ (with θ in radians).

The circular segment area is the region bounded by the chord and the arc: A = (r²/2) × (θ_rad − sin θ_rad). This equals the sector area (r²θ/2) minus the triangle area (r² sin θ/2). Inputs must satisfy r > 0 and 0° < θ < 360°; at exactly 0° the chord vanishes (zero length) and at exactly 360° it wraps around the full circle.