Frustum of a pyramid as a difference

A frustum of a pyramid is obtained by cutting a pyramid with a plane parallel to the base: its volume is the difference between the complete pyramid and the “small” one cut away.

Example — Sectioned pyramid = frustum

Frustum of a pyramid with an equilateral triangular base: side of the lower base $\ell=2$, side of the upper base $\ell'=1$, distance between the bases $h_{\text{tronco}}=3$.

Ratio of similarity: $k = \ell'/\ell = 1/2$. The heights of the “complete” pyramid and of the “small” one are in the ratio $h/h' = 1/k = 2$, with $h = h'+3$, from which $h' = 3$ and $h = 6$. $V_{\text{grande}} = \tfrac{1}{3}\cdot\tfrac{\sqrt{3}}{4}\cdot 4\cdot 6 = 2\sqrt{3}, \qquad V_{\text{piccola}} = \tfrac{1}{3}\cdot\tfrac{\sqrt{3}}{4}\cdot 1\cdot 3 = \tfrac{\sqrt{3}}{4}.$ $V_{\text{tronco}} = V_{\text{grande}} - V_{\text{piccola}} = 2\sqrt{3} - \tfrac{\sqrt{3}}{4} = \boxed{\tfrac{7\sqrt{3}}{4}}.$

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Topics: Synthetic geometry of space
Concepts: Similarity of solids · Frustum of a pyramid · Volume
Skills: Calculating