Prism Volume (General Cross-Section)

This is the parent formula that every other prism volume is a special case of. If the cross-section does not change along the length, the volume is simply cross-section times length, $V = BL$. Feed it $B = \pi r^2$ and you get the cylinder. Feed it $B = lw$ and you get the box. Feed it a triangle, an I-beam profile, a corrugated sheet, or the messy irregular outline of a river channel, and it still works.

The reason it works even for shapes that lean is Cavalieri's principle, named for Bonaventura Cavalieri, who argued in 1635 that two solids of the same height with equal cross-sections at every level have the same volume. That is why an oblique prism holds exactly as much as an upright one of the same base and height, and why a leaning stack of coins occupies the volume of a neat one.

The unit bookkeeping is where people slip. $B$ is an area and $L$ is a length, so a cross-section in square inches multiplied by a length in feet gives nothing meaningful until you reconcile them. The solver handles that conversion for you, but the habit of checking dimensions by hand is worth keeping.