To find the volume of a cone that has the same height and base radius as a cylinder, you need to multiply the volume of the cylinder by \(\frac{1}{3}\).
An informal argument for this can be made by comparing the two shapes. If you imagine filling the cone with a liquid, you’ll notice that you can fit three identical cones into a single cylinder. This is because the cone tapers to a point, while the cylinder maintains a uniform thickness throughout its height.
The volume of a cylinder is calculated using the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \(r\) is the base radius and \(h\) is the height.
Since the cone takes up one-third of the space of the cylinder when they share the same dimensions, the volume of the cone can be found with the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Thus, to find the volume of the cone, you indeed multiply the volume of the cylinder by \(\frac{1}{3}\).