jmc

Since the cone is tangent to all sides of the base of the prism, the base of the prism is a square. Furthermore, if the radius of the base of the cone is $r$, then the side length of the square is $2r$.

Let $h$ be the common height of the cone and the prism. Then the volume of the cone is $$\frac{1}{3}\pi r^{2}h,$$ and the volume of the prism is $(2r{)}^{2}h=4r^{2}h$, so the desired ratio is $$\frac{\frac{1}{3}\pi r^{2}h}{4r^{2}h}=\boxed{\frac{\pi }{12}}.$$