How does Brahmagupta’s formula find the area of a cyclic quadrilateral?

Brahmagupta’s formula is a significant result in Indian mathematics that provides the area of a cyclic quadrilateral—a four-sided polygon whose vertices all lie on a single circle—based solely on the lengths of its sides.1

If a cyclic quadrilateral has sides of lengths $a$, $b$, $c$, and $d$, its area $A$ is calculated using the formula:

$$A=\sqrt{(s-a)(s-b)(s-c)(s-d)}$$

In this expression, $s$ represents the semiperimeter of the quadrilateral, defined as:

$$s=\frac{a+b+c+d}{2}$$

This formula is a generalization of Heron’s formula, which calculates the area of a triangle using its side lengths. Brahmagupta, an accomplished ancient Indian astronomer and mathematician, presented this result in his seminal work, the Brahma-sphuta-siddhanta, where he also provided methods for determining the lengths of the quadrilateral's diagonals.12