Brahmagupta's formula


In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral given the lengths of the sides.

Formula

Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths,,, as
where, the semiperimeter, is defined to be
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as approaches zero, a cyclic quadrilateral converges into a cyclic triangle, and Brahmagupta's formula simplifies to Heron's formula.
If the semiperimeter is not used, Brahmagupta's formula is
Another equivalent version is

Proof

Trigonometric proof

Here the notations in the figure to the right are used. The area of the cyclic quadrilateral equals the sum of the areas of and :
But since is a cyclic quadrilateral,. Hence. Therefore,
Solving for common side, in and, the law of cosines gives
Substituting and rearranging, we have
Substituting this in the equation for the area,
The right-hand side is of the form and hence can be written as
which, upon rearranging the terms in the square brackets, yields
Introducing the semiperimeter,
Taking the square root, we get

Non-trigonometric proof

An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.

Extension to non-cyclic quadrilaterals

In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:
where is half the sum of any two opposite angles. This more general formula is known as Bretschneider's formula.
It is a property of cyclic quadrilaterals that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, is 90°, whence the term
giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is
where and are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.

Related theorems