Pyramid
Geometry & MeasurementA 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 (like a square or triangle) and triangular sides that all come together at one point on top, called the apex. The Egyptian pyramids are famous square pyramids.
Example
A square pyramid with a base of $6$ m $\times$ $6$ m and height $4$ m: $V = (1/3) \times 36 \times 4 = 48$ m$^3$. It has $5$ faces ($1$ square $+ 4$ triangles), $8$ edges, and $5$ vertices.
Key Insight
Three identical square pyramids can be assembled into a cube. This shows that a pyramid holds exactly $1/3$ as much as a prism with the same base and height, explaining where the $1/3$ in the volume formula comes from.
Definition
A pyramid has a polygonal base and triangular faces (called lateral faces) meeting at the apex. 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. The slant height $l$ is the height of a lateral triangular face.
Example
Regular square pyramid with base side $8$ m and lateral edge $10$ m. Height $h = \sqrt{10^2 - (4\sqrt{2})^2} = \sqrt{100 - 32} = \sqrt{68} = 8.25$ m. $V = (1/3)(64)(8.25) = 176$ m$^3$.
Key Insight
Any pyramid can be divided into tetrahedra (triangular pyramids). The tetrahedron is the simplest pyramid and the simplest 3-D solid; all pyramids and polyhedra can be decomposed into tetrahedra, making the tetrahedron the 3-D analog of the triangle.
Definition
A pyramid is the convex hull of a polygon base $P$ and an apex point $A$ not in the plane of $P$. Volume $V = (1/3)\text{Area}(P)h$ is derived by integration: $V = \int_0^h \text{Area}(\text{cross-section at } z) \, dz$. Since the cross-section at height $z$ is a scaled version of $P$ with scale factor $(1 - z/h)$, its area is $\text{Area}(P)(1-z/h)^2$, and integrating gives $(1/3)\text{Area}(P)h$.
Example
A frustum (truncated pyramid) with base area $B_1$, top area $B_2$, height $h$: $V = (h/3)(B_1 + B_2 + \sqrt{B_1 B_2})$. Setting $B_2 = 0$ gives the pyramid formula; setting $B_1 = B_2$ gives the prism formula, showing the frustum generalizes both.
Key Insight
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, a result that resolved Hilbert's third problem in 1900.