[2] Like many basic geometric terms, the word prism (from Greek πρίσμα (prisma) 'something sawed') was first used in Euclid's Elements.

Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”.

However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers).

A right prism (with rectangular sides) with regular n-gon bases has Schläfli symbol { }×{n}.

The volume of a prism whose base is an n-sided regular polygon with side length s is therefore:

The surface area of a right prism whose base is a regular n-sided polygon with side length s, and with height h, is therefore: The symmetry group of a right n-sided prism with regular base is Dnh of order 4n, except in the case of a cube, which has the larger symmetry group Oh of order 48, which has three versions of D4h as subgroups.

[7] A twisted prism is a nonconvex polyhedron constructed from a uniform n-prism with each side face bisected on the square diagonal, by twisting the top, usually by ⁠π/n⁠ radians (⁠180/n⁠ degrees) in the same direction, causing sides to be concave.

The simplest twisted prism has triangle bases and is called a Schönhardt polyhedron.

A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.

A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces.

A uniform star prism will have Schläfli symbol {p/q} × { }, with p rectangles and 2 {p/q} faces.

For a regular polygon base, the appearance is an n-gonal hour glass.

An n-gonal toroidal prism has 2n vertices, 2n faces: n squares and n crossed rectangles, and 4n edges.

The first examples of these exist in 4-dimensional space; they are called duoprisms as the product of two polygons in 4-dimensions.

For example, {4}×{4}, a 4-4 duoprism is a lower symmetry form of a tesseract, as is {4,3}×{ }, a cubic prism.