Volume Ratio Calculation for Cone and Cylinder

The problem asks for the ratio between the volume of a cone and a cylinder when they share the same height and base radius.

Given Information

• Let the height of both the cone and the cylinder be $h$.
• Let the radius of the base of both the cone and the cylinder be $r$.

Volume Formulas

Recall the standard formulas for the volume of a cone and a cylinder:

• Volume of a Cone ($V_{cone}$) = $\frac{1}{3}\pi r^2 h$
• Volume of a Cylinder ($V_{cylinder}$) = $\pi r^2 h$

Calculating the Ratio

We need to find the ratio $\frac{V_{cone}}{V_{cylinder}}$. Substitute the formulas:

Ratio = $\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h}$

Simplifying the Ratio

Cancel out the common terms ($\pi$, $r^2$, and $h$) from the numerator and the denominator:

Ratio = $\frac{\frac{1}{3}}{1} = \frac{1}{3}$

Therefore, the ratio of the volume of the cone to the volume of the cylinder is $1:3$.