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 $V_{cone}$ and $V_{prism}$ be the volume of a cone and of a prism, respectively. In a similar way, let $B_{cone}$ and $B_{prism}$ 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_{cone} = π 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_{prism}

can be as small or as large as we want.

V_{cone} V_{prism} = \frac{1}{3} B_{cone} h = B_{prism} h

Therefore, we can modify the base of the prism that

B_{prism} = \frac{1}{3} B_{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

B_{prism} \neq = \frac{1}{3} B_{cone},

the volumes of the cone and the prism would be different. Our answer is sometimes.

Exercise