In the triangle ABC call the incenter point P.
The radius r of the incircle is just the altitude from P to each side of the triangle. Let a,b,c be the sides opposite A,B,C, respectively.
Draw a line from P to each vertex, A,B,C. This divides the triangle into three smaller triangles, each with height r.
So, the area of ABC is just r/2 (a+b+c)
Or, where p is the perimeter of ABC:
A = r/2 * p
r = 2A/p
This is true for all triangles, including right triangles.