Nice question, help!

A right circular cone is inscribed in a right prism as shown. What is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of  pi.

Let the height of the prism = H  

Since the cone touches all four sides of the prism, the base must be square

So.....Let S be the side of the prism

So.....the volume of the prism is  S^2 *H  (1)

The radius of the cone  =  (1/2)S  and its height is H

So....its  volume  is   (1/3)*pi *(1/2 S)^2 * H   = (1/12)*pi*S^2*H  (2)

So.....the  ratio of (2)  to (1)  is

(1/12)* pi * S^2 * H                    pi

________________  =            ___

    S^2  * H                                12