Bretschneider's formula - Area of a general quadrilateral

Description

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

A quadrilateral is a polygon with four sides (or edges) and four vertices or corners. The interior angles of a simple (and planar) quadrilateral add up to 360 degrees of arc.

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.

Related formulas

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 e and f, as shown here:

Related formulas

Variables

Karea of the quadrilateral (m2)
ssemiperimeter (m)
aside a (m)
bside b (m)
cside c (m)
dside d (m)
αangle between sides a and d (degree)
γangle between sides b and c, opposite to α (degree)