Definition

The volume of a pyramid with base area $B$ and perpendicular height $h$ is $V = (1/3)Bh$, regardless of the base polygon's shape (triangle, square, hexagon, and so on) and holding for oblique pyramids too, where $h$ is still the perpendicular distance from apex to base plane. As an integral, $$V = \int_0^h B(1-z/h)^2 \, dz = Bh\int_0^1 (1-t)^2 \, dt = \frac{Bh}{3},$$ since the cross-section at height $z$ is a scaled copy of the base with linear scale factor $(1-z/h)$ and area $B(1-z/h)^2$, and integrating a quadratic function yields the $1/3$ factor. A frustum (pyramid with the top cut off) with base area $B_1$, top area $B_2$, and height $h$ has $V = (h/3)(B_1 + B_2 + \sqrt{B_1 B_2})$, a formula derived in ancient Egypt (Moscow Mathematical Papyrus, ca. 1850 BCE), one of the earliest known volume calculations.