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Let $P$ be a polygon.

Let $Q$ be a point not in the plane of $P$.

From each vertex of $P$, let lines be drawn to $Q$.

The polyhedron which is contained by $P$ and the triangles formed by the sides of $P$ and the lines to $Q$ is a pyramid.

In the words of Euclid:

A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point.

(The Elements: Book $\text{XI}$: Definition $12$)

Apex of Pyramid

The vertex of a pyramid which is the common vertex of its triangular faces is called the apex of the pyramid.

In the above diagram, $Q$ is the apex.

Base of Pyramid

The polygon of a pyramid to whose vertices the apex is joined is called the base of the pyramid.

In the above diagram, $ABCDE$ is the base of the pyramid $ABCDEQ$.

Height of Pyramid

The height of a pyramid is the length of the perpendicular from the plane of the base to its apex.

In the above diagram, $h$ is the height.

Also see

Linguistic Note

The word pyramid comes directly from the Classical Greek pyramis (πυραμίς) whose derivation is unknown.

In natural language the word usually refers to a square pyramid, and usually calls to mind the Great Pyramid of Giza.