In geometry, 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 as
where , the semiperimeter, is
The area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
Brahmagupta's formula may be seen as a formula in the half-lengths of the sides, but it also gives the area as a formula in the altitudes from the center to the sides, although if the quadrilateral does not contain the center, the altitude to the longest side must be taken as negative.
Proof of Brahmagupta's theorem
Area of the cyclic quadrilateral is
But since is a cyclic quadrilateral,
Applying law of cosines for and and equating the expressions for side , we have
Substituting (since angles 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 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 two opposite angles. The pair is irrelevant: if the other two angles are taken, half their sum is the supplement of . Since , we have .
where are the lengths of the diagonals of the quadrilateral.
It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to . Consequently, in the case of an inscribed quadrilateral, , whence the term
giving the basic form of Brahmagupta's formula.