In geometry, **Bretschneider's formula** is the following expression for the area of a quadrilateral,

Here, *p*, *q*, *r* and *s* are the sides of the quadrilateral, *T* is half the perimeter, and *A* and *C* are two opposite angles.

Bretschneider's formula works on any quadrilateral regardless of whether it is cyclic or not.

## Proof of Bretschneider's formula

Denote the area of the quadrilateral by *S*. Then we have

Therefore

The cosine law implies that

because both sides equal the square of the length of the diagonal *BD*. This can be rewritten as

Substituting this in the above formula for yields

This can be written as

Introducing the semiperimeter

the above becomes

and Bretschneider's formula follows.

## 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.

## External links

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