GeoProof — Brahmagupta's Formula explorer

Area = √( (s−a)(s−b)(s−c)(s−d) ) where s = semiperimeter = (a+b+c+d)/2

Drag the four vertices around the circle. The side-based formula always matches the directly measured area.

Brahmagupta's Formula

For a cyclic quadrilateral — one whose four vertices lie on a circle — with side lengths a, b, c, d and semiperimeter s = (a + b + c + d) / 2, the area is:

Area = √( (s−a)(s−b)(s−c)(s−d) )

Remarkably, the area depends only on the side lengths (for a cyclic quadrilateral), not on any angle.

It generalizes Heron's formula

Let one side shrink to zero (d → 0) and the quadrilateral becomes a triangle; the formula collapses to Heron's formula √(s(s−a)(s−b)(s−c)). So Heron is the triangle case of Brahmagupta.

Cyclic gives the maximum area

Among all quadrilaterals with the same four side lengths in the same order, the cyclic one has the largest possible area. So Brahmagupta's value is also the maximum area those four sides can enclose.

The general (non-cyclic) case

For a quadrilateral that isn't cyclic, Bretschneider's formula adds a correction term involving the two opposite angles; when those angles sum to 180° (the cyclic condition) the correction vanishes and Brahmagupta's formula returns.

A bit of history

Brahmagupta (598–668 CE), the Indian mathematician and astronomer, gave this result in his work the Brāhmasphuṭasiddhānta in 628 CE.