Problem #WSP-000336

Problem

We ‘typically’ use the formula $\frac{1}{2}bh$ for the area of a triangle, where $b$ is the length of the base, and $h$ is the perpendicular height. Here’s another one, called Heron’s formula.

Call the sides of the triangle $a$, $b$ and $c$. The perimeter is $a+b+c$. We call half of this the semiperimeter, $s=\frac{a+b+c}{2}$. Then the area of this triangle is $$\sqrt{s(s-a)(s-b)(s-c)}.$$ Prove this formula is correct.