# Derivation of Formula for Radius of Incircle

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

where A_{t} = area of the triangle and s = semi-perimeter.

**Derivation**

Let

A_{t} = Area of triangle ABC

A_{t} = Area of triangle BOC + Area of triangle AOC + Area of triangle AOB

$A_t = A_{BOC} + A_{AOC} + A_{AOB}$

$A_t = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr$

$A_t = \frac{1}{2}(a + b + c)\,r$

Let $\frac{1}{2}(a + b + c) = s$, the semi-perimeter

Thus,

$A_t = sr$

$r = \dfrac{A_t}{s}$