If you "unroll" the triangle by breaking it into three triangles, where the bases of the triangles are the original 3 sides, and the altitudes of the three triangles are all equal to the radius of the incenter, then the area is the sum of the 3 smaller triangles, and yet this area is _independent_ of everything but the perimeter and the radius of the incenter:

A = A_1 + A_2 + A_3 = p*r, where r = radius of the incircle

This should be (1/2) p * r, of course.