Heron's formula

In geometry, Heron's formula states that the area S of a triangle whose sides have lengths a, b, c is given by

where

Table of contents

1 History
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, is the following. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have

by the law of cosines. From this we get with some algebra

The altitude of the triangle on base a has length bsin(C), and it follows

Here the somewhat tedious but simple algebra in the last step was omitted.

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,

illustrates its similarity to Tartaglia's formula for the volume of a four-simplex.