# Heron's formula

{{#invoke:Hatnote|hatnote}}

In geometry, **Heron's formula** (sometimes called Hero's formula) is named after Hero of Alexandria^{[1]} and states that the area of a triangle whose sides have lengths *a*, *b*, and *c* is

where *s* is the semiperimeter of the triangle; that is,

Heron's formula can also be written as

Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.

## Example

Let ΔABC be the triangle with sides *a*=4, *b*=13 and *c*=15.
The semiperimeter is , and the area is

In this example, the side lengths and area are all integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or all of these numbers is not an integer.

## History

The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, *Metrica*, written *c.* A.D. 60. It has been suggested that Archimedes knew the formula over two centuries earlier,^{[2]} and since *Metrica* is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.^{[3]}

A formula equivalent to Heron's, namely

was discovered by the Chinese independently of the Greeks. It was published in *Shushu Jiuzhang* (“Mathematical Treatise in Nine Sections”), written by Qin Jiushao and published in A.D. 1247.

## Proofs

Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as below, or to the incenter and one excircle of the triangle [2].

### Trigonometric proof using the Law of cosines

A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows.^{[4]}
Let *a*, *b*, *c* be the sides of the triangle and *α*, *β*, *γ* the angles opposite those sides. We have

by the law of cosines. From this proof get the algebraic statement that

The altitude of the triangle on base *a* has length *b*·sin *γ*, and it follows