Ptolemy's Theorem Calculator

About this calculator

Ptolemy's theorem states that for any cyclic quadrilateral — one whose four vertices all lie on a single circle — the product of the diagonals equals the sum of the products of the two pairs of opposite sides: AC·BD = AB·CD + BC·DA. Given the four side lengths in order and one diagonal, this calculator rearranges the identity to solve for the missing diagonal, a relationship first recorded by the Greek astronomer Ptolemy in the Almagest to build his table of chords.

The theorem is a standard tool in Euclidean geometry courses and competition mathematics, where it's used to prove trigonometric identities (such as the sine addition formula), verify whether four points are concyclic, and solve olympiad-style problems involving inscribed polygons. It also generalizes into Ptolemy's inequality, which holds for any quadrilateral and reduces to equality only in the cyclic case — a fact geometry students and teachers use to test whether a given quadrilateral can be inscribed in a circle at all.