Ratio - volume of a pyramid $ABCF$ to volume of a cuboid $ABCDEFGH$

Question

Let's suppose we have a cuboid $ABCDEFGH$ and now we cut the pyramid $ABCF$ of that cuboid.

What is the ratio of volume of the pyramid to the volume of the original cuboid?

Is it $1\over4$ or $1\over3$?

Answer 1

Your subsequent comment indicates that you’re talking about a cuboid in the narrower sense, i.e., a right (or rectangular) cuboid. Let the base dimensions of the cuboid be $a$ and $b$, and let $c$ be its height. The pyramid has as base the $\triangle ABC$, whose area is $\frac12ab$, and height $c$. Its volume is one-third that of the corresponding cylinder, so its volume is $$\frac13\cdot\frac12abc=\frac16abc\;,$$ $\frac16$ the volume of the cuboid.

This image and the first one on this page show a decomposition of a cube into six pyramids of this type.