Brahmagupta's Formula Calculator
Area of a cyclic quadrilateral from four side lengths using Brahmagupta formula A = sqrt((s-a)(s-b)(s-c)(s-d)).
Heron generalized to four sides.
Brahmagupta’s formula gives the area of a cyclic quadrilateral (a four-sided shape whose four corners all sit on the same circle) using only the four side lengths.
A = √((s − a)(s − b)(s − c)(s − d)), where s = (a + b + c + d) / 2 is the semi-perimeter.
Brahmagupta wrote this down in India in 628 CE, in his book Brahmasphutasiddhanta. It is one of the oldest closed-form area formulas in geometry, and it has the same structure as Heron’s formula for triangles. In fact, if you set d = 0, the quadrilateral collapses into a triangle and the formula becomes √(s(s − a)(s − b)(s − c)). That is Heron, exactly.
Which shapes are cyclic? Every rectangle is cyclic (its diagonals are equal and meet at the centre of the circumscribed circle). Every square is cyclic. Every isosceles trapezoid is cyclic. A general trapezoid or kite is not, unless its angles happen to land just right. The rule: a quadrilateral is cyclic if and only if its opposite angles sum to 180°.
What if my shape isn’t cyclic? Then Brahmagupta gives you an upper bound, not the true area. Among all quadrilaterals with the four given side lengths, the cyclic one has the largest possible area. A “floppy” quadrilateral (think of four sticks loosely pinned at the corners) sweeps through many shapes; the moment they happen to be concyclic, area is maximum. Bretschneider’s formula extends Brahmagupta to handle the general case by subtracting a correction term that depends on the opposite angles.
Worked example — Sides 5, 7, 8, 10. Semi-perimeter s = (5 + 7 + 8 + 10) / 2 = 15. Then (s − a)(s − b)(s − c)(s − d) = 10 × 8 × 7 × 5 = 2800. Area = √2800 ≈ 52.92 square units. Without the cyclic assumption you can’t get a single number for the area — you’d need at least one angle.
Real uses: survey work where four boundary lengths are known and the plot is approximately cyclic; mediaeval Indian astronomy (Brahmagupta’s original motivation was lunar geometry); competition geometry problems; the fact that any rectangle’s area can be computed two ways gives a quick sanity check on a measurement.
Sanity check the inputs: the four side lengths must satisfy a “quadrilateral inequality” — the sum of any three sides must exceed the fourth. Otherwise the sides cannot close into a polygon at all, and the formula returns an imaginary number.