Heron's formula




In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria,^[1] gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.

Example

Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. The semiperimeter is s = 1/2(a + b + c) = 1/2(4 + 13 + 15) = 16, and the area is

${\displaystyle {\begin{aligned}A&={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\sqrt {16\cdot (16-4)\cdot (16-13)\cdot (16-15)}}\\&={\sqrt {16\cdot 12\cdot 3\cdot 1}}={\sqrt {576}}=24.\end{aligned}}}$

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.