We are asked to determine whether the following statement is always, sometimes, or never true.
The volume of a cone with radius r and height h equals the volume of a prism with height h.
Let V_\text{cone} and V_\text{prism} be the volume of a cone and of a prism, respectively. In a similar way, let B_\text{cone} and B_\text{prism} be the area of the base of the cone and of the prism.
Since the base of the cone is a circle with radius r, its area is B_\text{cone}=\pi r^2. Notice that we do not know anything about the base of the prism. It can be a quadrangle, it can be a pentagon, it can be small, it can be large. This tells us that B_\text{prism} can be as small or as large as we want.
\begin{aligned}
V_\text{cone}&=\textcolor{darkorange}{\dfrac{1}{3}B_\text{cone}}h \\
V_\text{prism}&=\textcolor{darkorange}{B_\text{prism}} h
\end{aligned}
Therefore, we can modify the base of the prism that \textcolor{darkorange}{B_\text{prism}}=\textcolor{darkorange}{\frac{1}{3}B_\text{cone}}. The volume of the cone and the volume of the prism would be equal. But if we modify the base of the prism that \textcolor{darkorange}{B_\text{prism}}\neq\textcolor{darkorange}{\frac{1}{3}B_\text{cone}}, the volumes of the cone and the prism would be different. Our answer is sometimes.
