Octagonal prism

The octagonal prism, or op, is a prismatic uniform polyhedron. It consists of 2 octagons and 8 squares. Each vertex joins one octagon and two squares. As the name suggests, it is a prism based on an octagon.

It can also be obtained from the small rhombicuboctahedron by removing two opposing square cupolas. It can therefore also be thought of as a bidiminished small rhombicuboctahedron.

Vertex coordinates
An octagonal prism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}2,\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt2}2,\,±\frac12,\,±\frac12\right).$$

Representations
An octagonal prism has the following Coxeter diagrams:


 * x x8o (full symmetry)
 * x x4x (generally a ditetragonal prism)
 * s2s8x (generally a ditetragonal trapezoprism)
 * xx8oo&#x (octagonal frustum)
 * xx4xx&#x (ditetragonal frustum)
 * xxxx xwwx&#xt (A1×A1 axial)
 * xx xw wx&#zx (A1×A1×A1 symmetry)

Semi-uniform variant
The octagonal prism has a semi-uniform variant of the form x y8o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by $$\sqrt{a^2\frac{2+\sqrt2}{2}+\frac{b^2}{4}}$$ and its volume is given by $$2(1+\sqrt2)a^2b$$.

An octagonal prism with base edges of length a and side edges of length b can be alternated to form a square antiprism with base edges of length $$\sqrt{2+\sqrt2}a$$ and side edges of lengths $$\sqrt{a^2+b^2}$$. In particular if the side edges are $$\sqrt{1+\sqrt2}$$ times the length of the base edges this gives a uniform square antiprism.

Variations
An octagonal prism has the following variations:


 * Ditetragonal prism - prism with ditetragons as bases, and 2 types of rectangles
 * Ditetragonal trapezoprism - isogonal with trapezoid sides
 * Octagonal frustum
 * Ditetragonal frustum

Related polyhedra
A square cupola can be attached to a base of the octagonal prism to form the elongated square cupola. If a second square cupola is attached to the other base in the same orientation, the result is the elongated square orthobicupola, better known as the small rhombicuboctahedron. If the second cupola is rotated 45º the result is the elongated square gyrobicupola.

The small rhombihexahedron can be obtained by blending three orthogonal octagonal prisms so that half of the square faces blend out.