Brahmagupta's Formula

For a general quadrilateral with sides of length , , , and , the area is given by

(1)

where

(2)

is the semiperimeter, is the angle between and , and is the angle between and . Brahmagupta's formula

(3)

is a special case giving the area of a cyclic quadrilateral (i.e., a quadrilateral inscribed in a circle), for which . In terms of the circumradius of a cyclic quadrilateral,

(4)

The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.

For a bicentric quadrilateral (i.e., a quadrilateral that can be inscribed in one circle and circumscribed on another), the area formula simplifies to

			(5)
			(6)

(Ivanoff 1960; Beyer 1987, p. 124).