Chord Length Calculator

Chord Length Calculator: Formula, Methods, and Applications

A chord is a straight line segment whose two endpoints both lie on a circle's circumference. The central angle subtended by those endpoints, the arc length between them, and the sagitta (the perpendicular rise from chord midpoint to arc) all define the same geometric relationship from different perspectives. This calculator determines chord length from any of the three input pairs, covering the full range of real-world measurement scenarios.

Method 1: Radius and Central Angle

The primary chord length formula derives from basic trigonometry. Drawing two radii of length r to the chord endpoints and bisecting the central angle θ produces two congruent right triangles. In each triangle the hypotenuse is r and the side opposite the half-angle is c/2. Applying the sine definition gives the standard result:

c = 2r ⋅ sin(θ / 2)

For a circle with radius 10 m and a central angle of 60° (π/3 rad), the chord is 2 × 10 × sin(30°) = 2 × 10 × 0.5 = 10 m. At 90° the chord becomes 2 × 10 × sin(45°) ≈ 14.14 m, and at 180° it equals the diameter, 20 m. This formula is specified in highway surveying practice by the Federal Highway Administration Field Formulas (M 22-24) for staking circular road curves, and its geometric validity is confirmed in orthopedic research published at PMC — Mathematical Modeling of Glenoid Bone Loss, where surgeons measure shoulder-socket defects using chord geometry against an ideal reference circle.

Method 2: Radius and Arc Length

When arc length s is the known quantity rather than the central angle, the radian arc-length relationship s = rθ allows direct substitution. Solving for θ yields θ = s/r (in radians), which feeds into the primary formula:

c = 2r ⋅ sin(s / (2r))

Example: for a radius of 8 m and an arc length of 6 m, θ = 6/8 = 0.75 rad, and c = 2 × 8 × sin(0.375) ≈ 2 × 8 × 0.3664 ≈ 5.86 m. The arc-length calculus foundation underlying this conversion is detailed by Paul's Online Math Notes — Arc Length.

Method 3: Radius and Sagitta

The sagitta (height, denoted h) is the perpendicular distance from the midpoint of the chord to the arc. Applying the Pythagorean theorem to the right triangle formed by radius r, half-chord c/2, and the remaining radial segment (r − h):

(c/2)² = r² − (r − h)² = 2rh − h²

Solving for chord length: c = 2√(h(2r − h))

For r = 15 m and h = 3 m: c = 2√(3 × 27) = 2√81 = 18 m. The sagitta method is preferred in arch construction, curved cabinetry, and road-camber measurement because the rise is directly measurable with a straightedge and ruler without locating the circle's center.

Angle Unit Conversion

The sine formula requires θ in radians. Degree inputs convert via θ_rad = θ_deg × π / 180. Gradian inputs convert via θ_rad = θ_grad × π / 200. Selecting the wrong unit produces dramatically incorrect results — a 60-degree chord and a 60-radian chord on the same circle differ by orders of magnitude, so always confirm the angle unit before computing.

Real-World Applications

• Road and rail surveying: The FHWA uses chord-offset staking to lay out horizontal curves, converting arc stations to ground-measurable chord distances.
• Architecture and structural engineering: Arched roofs, barrel vaults, and curved façades require chord and sagitta values to specify material cut lengths and formwork geometry.
• Orthopedic surgery: Surgeons quantify glenoid bone loss by measuring the chord across the defect and expressing the deficit as a percentage of the reference circle's diameter.
• CNC woodworking and fabrication: Curved panels, custom trim, and pipe bends all use chord and sagitta measurements to program tool paths or mark manual cuts.
• Optics and telescope making: Mirror and lens sagitta depth relative to the aperture chord defines the surface curvature tolerance in precision optical instruments.