Bretschneider's formula


In geometry, Bretschneider's formula is the following expression for the area of a general quadrilateral:
Here,,,, are the sides of the quadrilateral, is the semiperimeter, and and are two opposite angles.
Bretschneider's formula works on any quadrilateral, whether it is cyclic or not.
The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt.

Proof

Denote the area of the quadrilateral by. Then we have
Therefore
The law of cosines implies that
because both sides equal the square of the length of the diagonal. This can be rewritten as
Adding this to the above formula for yields
Note that:
Following the same steps as in Brahmagupta's formula, this can be written as
Introducing the semiperimeter
the above becomes
and Bretschneider's formula follows after taking the square root of both sides:

Related formulae

Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.
The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals and to give