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 \(\pi\) .

tertre Mar 21, 2018

#2**+1 **

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

CPhill Mar 21, 2018