Visual Proof: Where the Volume of a Pyramid Formula Actually Comes From

An essential feature on Visual Proof: Where the Volume of a Pyramid Formula Actually Comes From, featuring in-depth facts.

The cube dissection neatly proves the square pyramid case, but pyramids come in diverse geometries: the triangular pyramid, the rectangular pyramid, and irregular polygonal forms. Proving that the one-third relationship governs all of them relies on Italian mathematician Bonaventura Cavalieri's foundational seventeenth-century geometric insights.

Cavalieri's Principle establishes that if two three-dimensional solids share identical total heights, and every horizontal cross-sectional area taken at equal elevations from the base is equal, both solids must contain identical total volumes.

Consider a pyramid of height $h$ and base area $B$. Slice through the pyramid horizontally at a distance $y$ below the apex. Because the linear dimensions of any cross section scale directly with distance from the tip, the linear scale factor at depth $y$ is $\frac{y}{h}$.

Area scales quadratically relative to linear dimensions. The cross-sectional area $A(y)$ at elevation $y$ is governed by a simple square ratio:

$$A(y) = B \left(\frac{y}{h}\right)^2 = \frac{B}{h^2} y^2$$

This quadratic scaling profile depends solely on perpendicular depth and total base area. The perimeter shape does not alter this equation. A triangular pyramid, a rectangular pyramid, and an oblique, tilted cone with matching base area and perpendicular height will feature identical cross-sectional areas at every single slice. Because their thin horizontal layers match from base to tip, their cumulative capacities match. The one-third multiplier applies universally across all pyramid styles.