Cone and Cylinder Volume Ratio

Consider a right circular cone with a base radius of $R$ and a height of $H$. If the cone is inscribed within a right circular cylinder that has the same base radius $R$ and height $H$, what is the ratio of the volume of the cone to the volume of the cylinder?

Recall that the volume $V$ of a right circular cone can be calculated using the formula:

$V =\frac{1}{3} imes ext{Base Area} imes ext{Height} =\frac{1}{3} imes ext{Area of Circle} imes H =\frac{1}{3} imes op imes R^2 imes H$,

while the volume of a cylinder is given by:

$V = ext{Base Area} imes ext{Height} = ext{Area of Circle} imes H = op imes R^2 imes H$.