Pyramid
A pyramid is a 3-D solid with a polygon base and triangular faces that meet at a single apex point.
Formula
V = \frac{1}{3} \times B \times h
Definition
A pyramid has a flat polygon base with triangular sides that all meet at a single apex point, like the Egyptian pyramids; a right pyramid has its apex directly above the centroid of the base. Volume $V = (1/3)Bh$ where $B$ is the base area and $h$ is the perpendicular height, regardless of the base polygon's shape, and the slant height $l$ is the height of a lateral triangular face. Formally, a pyramid is the convex hull of a polygon base $P$ and an apex point $A$ not in the plane of $P$; since the cross-section at height $z$ is a scaled copy of $P$ with factor $(1 - z/h)$ and area $\text{Area}(P)(1-z/h)^2$, integrating gives $V = \int_0^h \text{Area}(\text{cross-section at } z) \, dz = (1/3)\text{Area}(P)h$.
Example
A square pyramid with base $6$ m $\times$ $6$ m and height $4$ m has $V = (1/3) \times 36 \times 4 = 48$ m$^3$, with $5$ faces ($1$ square $+ 4$ triangles), $8$ edges, and $5$ vertices. A regular square pyramid with base side $8$ m and lateral edge $10$ m has height $h = \sqrt{10^2 - (4\sqrt{2})^2} = \sqrt{68} = 8.25$ m, giving $V = (1/3)(64)(8.25) = 176$ m$^3$. A frustum (truncated pyramid) with base area $B_1$, top area $B_2$, height $h$ has $V = (h/3)(B_1 + B_2 + \sqrt{B_1 B_2})$; setting $B_2 = 0$ recovers the pyramid formula, and setting $B_1 = B_2$ recovers the prism formula.
Key Insight
Three identical square pyramids can be assembled into a cube, showing that a pyramid holds exactly $1/3$ as much as a prism with the same base and height, where the $1/3$ in the formula comes from. Any pyramid can be divided into tetrahedra (triangular pyramids), the simplest pyramid and the simplest 3-D solid, into which all pyramids and polyhedra can be decomposed. The Dehn invariant shows that not all solids with equal volume can be dissected into each other using finitely many polyhedral pieces, a cube and a regular tetrahedron of equal volume cannot be so related, resolving Hilbert's third problem in 1900.