Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?



Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum
Let the pyramid $P$ have volume $V\_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 \leq h \leq H$. You want to know the volume of the frustum. Call it $V\_F$.


By calculus, $V\_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid above the frustum is a scaled version of the original pyramid. It has height $H-h$, so the smaller pyramid is a scaled-down version of the original pyramid by the factor $(H-h)/H = 1 - h/H$. In particular, the base of the smaller pyramid has area $(1-h/H)^2A$, so the volume of the smaller pyramid is $(1/3)(1-h/H)^2A(H-h)$. Therefore
$$
V\_F = V\_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2A(H-h).
$$
Replacing $A$ by $3V\_P/H$,
$$
V\_F = V\_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2\frac{3V\_P}{H}(H-h) = \left(1 - \left(1 - \frac{h}{H}\right)^3\right)V\_P.
$$
So there is your frustum volume formula in terms of $V\_P$, $H$, and $h$. As a reality check, when $h = H$ we get $V\_F = V\_P$ (and the frustum $F$ in this case is the full pyramid $P$) and when $h = 0$ we get $0$ (and the frustum $F$ in this case is the base of the pyramid, so it is planar).


PS: Until less than two weeks ago, I always thought the term in English was *frustrum*. To learn so late in life that it is actually frustum was quite… frustating.


