Cone Frustum Volume (Truncated Cone)

A frustum is a cone with its tip sliced off parallel to the base, and it is the shape of almost every real container that tapers: buckets, plant pots, drinking cups, lampshades, silo discharge cones, and the classic paper snow-cone once you truncate it. The volume is $V = \tfrac{\pi h}{3}(R^2 + Rr + r^2)$, which is the big cone minus the little cone you removed, tidied up so that the vanished apex height never appears.

This is one of the oldest formulas humanity has written down. The Moscow Mathematical Papyrus, copied around 1850 BC, contains a worked problem for the volume of a truncated square pyramid using exactly the same $a^2 + ab + b^2$ structure, some fourteen centuries before Euclid. Whoever discovered it left no proof, only a confident recipe and the answer.

The pitfall is the tempting shortcut of averaging the radii and treating the shape as a cylinder. That gives $\left(\tfrac{R+r}{2}\right)^2$ where the truth is $\tfrac{1}{3}(R^2 + Rr + r^2)$, and for a bucket 2 m across the top and 1 m across the bottom those are 2.25 and 2.33 in units of $r^2$. The average always underestimates, by roughly four percent here and more as the taper steepens. Two reassuring checks live inside the real formula: set $R = r$ and it collapses to the cylinder $\pi r^2 h$, and set $r = 0$ and it collapses to the cone $\tfrac{1}{3}\pi R^2 h$.