Chord length calculator

When the distance from the center exceeds the radius the line misses the circle entirely, so the tool returns invalid_input instead of a complex length or a silent NaN. Through the API a successful request costs $0.002; validation failures are not charged.

What a chord length is and what the response contains

A chord is the straight line segment joining two points on a circle. Its length depends only on the radius and how far those points sit angularly or how far the chord line sits from the center: two points that subtend a larger central angle sit farther apart along the straight shortcut, and a chord that passes closer to the center is longer, reaching a maximum equal to the diameter when the distance is zero. This tool exists to turn those two standard textbook setups into a single deterministic response without forcing you to draw an isosceles triangle, bisect the angle by hand, or remember which form of the identity matches your given data. If you need arc length, sector area, or segment area as the primary answer, use a tool dedicated to those problems; the search intent here is the straight chord length and a clear answer or an honest error. Preferred inputs are the scalar radius plus either central_angle or distance as JSON numbers; decimal strings are accepted for form and spreadsheet callers. In angle mode the response reports mode angle, the echoed radius and central angle, angle_unit, central_angle_rad, chord_length, half_chord, sagitta for the minor-arc sense, distance_to_chord recovered from the half-chord Pythagorean relation, and a formula field recording the sine identity. In distance mode the response reports mode distance, the echoed radius and distance, chord_length, half_chord, sagitta as radius minus distance, central_angle_deg and central_angle_rad recovered via two arcsine of half-chord over radius, and the square-root formula. Missing radius, non-finite values, a non-positive radius, both modes supplied at once, neither mode supplied, a negative distance, or a distance that exceeds the radius all raise invalid_input with a message that names the failing condition. That discipline keeps automated tests stable, makes caching safe, and prevents a student or a script from treating a missing line as a real chord.

How the two modes are read and how the closed forms are evaluated

Exactly one of those two optional families must be present: supplying both is rejected so a caller cannot silently mix contradictory geometry, and supplying neither is rejected with a message that names the two valid modes. The central angle is converted to radians when needed, half of that angle is taken, and the chord length is two times the radius times the absolute value of the sine of the half-angle, which keeps length non-negative for positive or negative angle inputs. Companion fields fold the absolute angle into a minor arc on zero through pi so sagitta equals radius times one minus cosine of half that minor angle, and the perpendicular distance from the center to the chord is recovered as the square root of radius squared minus half-chord squared. Otherwise the chord is two times the square root of the non-negative radicand radius squared minus distance squared, sagitta is radius minus distance, and the central angle is recovered as two times arcsine of half-chord over radius, reported in both degrees and radians. Optional precision only rounds the published fields; the internal solve always uses full double precision before that final rounding step. The engine resolves radius from the field radius with aliases r and R, requiring a finite value strictly greater than zero. It then looks for central_angle with aliases angle, theta, and the theta symbol, or for distance with aliases d, dist, and offset. For angle mode the optional angle_unit field, with aliases unit and angleUnit, defaults to degrees; values deg, degree, degrees, or d select degrees, while rad, radian, radians, or r select radians. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the half-angle or Pythagorean identities yourself.

Where chord length fits in study, design, and code

On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other free geometry endpoints when you need circle construction from three points, line intersections, or triangle solvers for a lab of deterministic tools, and with documentation that shows both the radii and the finished chord so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for a chord length calculator or for the chord from radius and central angle, the contract is the same: radius plus angle or distance in, a finite chord or a clear invalid_input out.