Derivation of Formula for Radius of Incircle

The radius of incircle is given by the formula
 

r=Ats

 

where At = area of the triangle and s = semi-perimeter.
 

Derivation
 

Figure for derivation of radius of incircle

 

Let
At = Area of triangle ABC
At = Area of triangle BOC + Area of triangle AOC + Area of triangle AOB
 

At=ABOC+AAOC+AAOB

At=12ar+12br+12cr

At=12(a+b+c)r
 

Let   12(a+b+c)=s,   the semi-perimeter
 

Thus,
At=sr

r=Ats