, Brahmagupta's formula finds the area of any quadrilateral
given the lengths of the sides and some of their angles. In its
most common form, it yields the area of quadrilaterals that can
be inscribed in a circle In its basic and easiest-to-remember
form, Brahmagupta's formula gives the area of a cyclic
quadrilateral whose sides have lengths a, b, c, d as where s,
the semiperimeter, is determined by This formula generalizes
Heron's formula for the area of a triangle. The area of a cyclic
quadrilateral is the maximum possible area for any quadrilateral
with the given side lengths. Proof of Brahmagupta's formula Area
of the cyclic quadrilateral = Area of + Area of But since is a
cyclic quadrilateral, Hence Therefore Applying law of cosines
for and and equating the expressions for side we have
Substituting (since angles and are supplementary) and
rearranging, we have Substituting this in the equation for area,
which is of the form and hence can be written in the form as
Taking square root, we get Extension to no