Hero's formula

Hero's formula is an important formula (also known as Heron's formula) in plane geometry that allows the area of any triangle to be calculated without knowing the altitude (perpendicular height) of any of its sides. Let a, b, and c be the side lengths of a triangle and A its area. Hero's formula states that


A2 = s(s - a)(s - b)(s - c)


where s = (a + b + c)/2.


The origin of this formula is historically obscure. A medieval Arab source, for example, ascribes it to Archimedes. However, the first definite reference we have to it is by Hero of Alexandria. His proof is extremely convoluted, and it seems clear that it must have been arrived at by an entirely different thought process, and then dressed up in the usual synthetic form that the classical Greeks preferred for their presentations. Hero's formula contains Pythagoras's theorem as a degenerate case. A Heroian triangle is one with integer sides and integer area.