# Heron’s Formula

## Heron’s Formula

Formula that expresses the relationship between the side lengths of a triangle and its area.

### Formula

Heron’s formula can be used to calculate the area of a triangle when its three side lengths are given.

Given a triangle with side lengths a, b and c and a semiperimeter s, the area A of the triangle is calculated as follows:

$$A=\sqrt{p(p-a)(p-b)(p-c)}$$

### Example

Given a right triangle whose three sides measure 3 cm, 4 cm and 5 cm and whose semiperimeter is therefore 6, then Heron’s formula gives:

$$A=\sqrt{6(6-3)(6-4)(6-5)}=\sqrt{6(3)(2)(1)}=\sqrt{36}=6$$

### Historical note

Heron’s formula is credited to Heron of Alexandria (1st century BCE), who was an engineer and a mathematician also known for his work in mechanics. Many of his works were translated into Latin by European scholars (before the Middle Ages) and into Arabic following the advent of Islam in the region. In Book I of his work, Metrica, he provides methods for calculating the area of various plane geometric figures and solids and proposes the formula presented here to calculate the area of a triangle. Although this formula has been credited to Heron of Alexandria and bears his name, historians agree that it was developed by Archimedes.