Bretschneider's formula

In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral.

The formula also works on crossed quadrilaterals provided that directed angles are used.

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.

Bretschneider's formula is expressed as: Here, a, b, c, d are the sides of the quadrilateral, s is the semiperimeter, and α and γ are any two opposite angles, since

Denote the area of the quadrilateral by K. Then we have Therefore The law of cosines implies that because both sides equal the square of the length of the diagonal BD.

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 e and f to give[2][3]