# Bretschneider's Formula for The Area of Any Quadrilateral *Bretschneider's Formula* is a generalization of the [[Brahmagupta's Formula For The Area of Any Cyclic Quadrilateral]], which itself is a generalization of the [[Heron's Formula For The Area of Any Triangle]]. It is **used to calculate the area of any quadrilateral**. Let $s$ be the semi-perimeter of the quadrilateral ($s=\frac{a+b+c+d}{2}$), $\alpha$ and $\beta$ any two opposing angles, and $j$ and $k$ the lengths of the qudrilateral's two diagonals. The quadrilateral's area $A$ is then $ \begin{align} A&=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd \cos^2{\left(\frac{\alpha+\beta}{2}\right)}} \\ &=\sqrt{(s-a)(s-b)(s-c)(s-d)-\frac{1}{4}(ac+bd+jk)(ac+bd-jk)} \end{align} $ If the quadrilateral is also cyclic[^1], a simplified version of this formula can be used instead. It is called the [[Brahmagupta's Formula For The Area of Any Cyclic Quadrilateral]]. [^1]: A cyclic quadrilateral is a quadrilateral whose points all lie on a single [[circle]] i.e. one for which a circumcircle exists.