Prism Volume Theorem

First consider a right prism, whose lateral edges are perpendicular to the base. Slicing it with a plane parallel to the base at any height produces a cross-section congruent to the base itself, of area $B$. Stacking these cross-sections from height $0$ to $h$ gives volume $\int_0^h B\,dy = Bh$, which is exactly $V = Bh$ for the right prism.

Now take any oblique prism with the same base area $B$ and the same height $h$. Place it next to a right prism with that same base and height, sharing the plane of one base. A cross-section of the oblique prism at height $y$ is a translated copy of the base (translation does not change area), so it has area $B$, exactly the same as the cross-section of the right prism at that height.

By Cavalieri's principle, since the two solids have equal cross-sectional area $B$ at every height $y \in [0, h]$, they have equal volume. Since the right prism has volume $V = Bh$, so does the oblique prism, proving the formula for every prism.