Length of Inradius of Triangle

Theorem
Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$.

Then the length of the inradius $r$ of $\triangle ABC$ is given by:
 * $r = \dfrac {\sqrt {s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}} s$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

Proof
Let $\mathcal A$ be the area of $\triangle ABC$.

From Area of Triangle in Terms of Inradius:
 * $\mathcal A = r s$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

From Heron's Formula:
 * $\mathcal A = \sqrt {s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

Hence the result:
 * $r = \dfrac {\sqrt{s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)} } s$