Brahmagupta's formula is a special case of Bretschneider's formula as applied to cyclic quadrilaterals.
Bretschneider's formula states that the area of a quadrilateral is given by
where is the area of the quadrilateral, and are the angles, and are the sides of the quadrilateral, and is the semiperimeter of the quadrilateral, given by .
For a cyclic quadrilateral, we know that the sum of the opposite angles is . Hence we have . So the simplified version known as Brahmagupta's formula is given as follows:
Given a cyclic quadrilateral with side lengths , the area can be found as
Alternatively, without using the semiperimeter, we can use
Assume to be the vertices of the cyclic quadrilateral, and the lengths of sides respectively. Then
Since the quadrilateral is cyclic, so
Solving for common side in triangles and and using laws of cosines, we get
As angles and are supplementary, and
Substituting this into the equation of area we get
The RHS of the equation is of the form , so it can be written as
So, the area of the cyclic quadrilateral with sides can be written as