# Heron's formula

## Book Suggestions: |

**Heron's formula**states that the area

*S*of a triangle whose sides have lengths

*a*,

*b*,

*c*is given by

Table of contents |

2 Proof 3 Generalizations 4 See also |

## History

The formula is credited to Heron of Alexandria in the 1st century A.D., and a proof can be found in his book *Metrica*. It is now believed that Archimedes already knew the formula, and it is of course possible that it has been known long before.

## Proof

A modern proof, which uses algebra and trigonometry and is quite unlike the one provided by Heron, follows. Let *a*, *b*, *c* be the sides of the triangle and *A*, *B*, *C* the angles opposite those sides. We have

- .

*a*has length

*b*sin(C), and it follows

## Generalizations

The formula is in fact a special case of Brahmagupta's formula for the area of a cyclic quadrilateral; both of which are special cases of Bretschneider's formula for the area of a quadrilateral.

Expressing Heron's formula with a determinant in terms of the squares of the distances between the three given vertices,