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:
 * $r = \dfrac {\sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } } s$

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

Proof

 * Incircle.png

Let $\AA$ be the area of $\triangle ABC$.

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

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

From Heron's Formula:
 * $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

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

Hence the result:
 * $r = \dfrac {\sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } } s$