Equivalent pyramids via Cavalieri

Problem

Show, using Cavalieri’s principle, that two pyramids with bases of equal area and the same height have the same volume, even if the bases have different shapes.

Solution

Place the two pyramids with their bases on the same plane and their apexes at the same height $H$. At height $z$ (measured from the tip) the cross-section of each pyramid is similar to its own base with linear scale factor $z/H$, so its area is $A(z) = B\cdot\left(\frac{z}{H}\right)^2.$ If the two bases have the same area $B$ and the pyramids have the same height $H$, then at every height $z$ the two cross-sections have the same area $A(z)$. By Cavalieri’s principle (3D version), the two solids have the same volume: $\boxed{V_1 = V_2 = \tfrac{1}{3}B H.}$ The shape of the bases is irrelevant: all that matters is that the corresponding cross-sections have equal area.

Links

Topics: Integral
Concepts: Cavalieri’s principle · Volume by cross-sections
Methods: Cavalieri’s principle
Skills: Prove · Synthetic geometry
Exercise types: Proof