Inscribe a circle and label its radius, r.

What is the ratio of the area of the circle to that of the triangle?

Formula for r as a function of the sides of the triangle:
Picture the triangle plus the point at the center of the circle.  Draw 3 lines from the center of the circle, one to each vertex.  Now the triangle is divided into 3 smaller triangles and like ancient Gaul, the 3 parts add up to the whole.

(1/2)ab = (1/2)ra + (1/2)rb + (1/2)rh
ab= r*(a+b+h)
r= ab/(a+b+h)

From np_rt, Ratio = Pi*r/(r+h)
so

Ratio= Pi*a*b/(ab+ah+bh+h^2)

for an isosceles right triangle, this = .5390 (the maximum)
for a 3,4,5 triangle it is = .5236
for a 30-60-90 triangle it is = .4860