Brahmagupta's formula

In Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.

Brahmagupta's formula gives the area $K$ of a cyclic quadrilateral whose sides have lengths $a$ , $b$ , $c$ , $d$ as

where $s$ , 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 $d$ approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.

If the semiperimeter is not used, Brahmagupta's formula is

Another equivalent version is

Here the notations in the figure to the right are used. The area $K$ of the cyclic quadrilateral equals the sum of the areas of $△ ADB$ and $△ BDC$ :

But since $ABCD$ is a cyclic quadrilateral, $∠ DAB = 180° - ∠ DCB$ . Hence $sin A = sin C$ . Therefore,

Solving for common side $DB$ , in $△ ADB$ and $△ BDC$ , the law of cosines gives

Substituting $cos C = -cos A$ (since angles $A$ and $C$ are supplementary) and rearranging, we have

Substituting this in the equation for the area,

The right-hand side is of the form $a 2 - b 2 = (a - b)(a + b)$ and hence can be written as

which, upon rearranging the terms in the square brackets, yields

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