Decagonal prism

The decagonal prism, or dip, is a prismatic uniform polyhedron. It consists of 2 decagons and 10 squares. Each vertex joins one decagon and two squares. As the name suggests, it is a prism based on a decagon.

It is the highest convex polygonal prism to occur as cells in uniform polychora.

Vertex coordinates
A decagonal prism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right).$$

Representations
A decagonal prism has the following Coxeter diagrams:


 * x x10o (full symmetry)
 * x x5x (as dipentagonal prism)
 * s2s10x (as dipentagonal trapezoprism)
 * xx10oo&#x (decagonal frustum)
 * xx5xx&#x (dipentagonal frustum)
 * xxxxx xFVFx&#xt (A1×A1 axial, square-first)

Semi-uniform variant
The decagonal prism has a semi-uniform variant of the form x y10o 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 $$\sqrt{a^2\frac{3+\sqrt5}{2}+\frac{b^2}{4}}$$ and its volume is given by $$5\frac{\sqrt{5+2\sqrt5}}{2}a^2b$$.

A decagonal prism with base edges of length a and side edges of length b can be alternated to form a pentagonal antiprism with base edges of length $$\sqrt{\frac{5+\sqrt5}{2}}a$$ and side edges of lengths $$\sqrt{a^2+b^2}$$. In particular if the side edges are $$\frac{1+\sqrt5}{2}$$ times the length of the base edges this gives a uniform pentagonal antiprism.

Variations
A decagonal prism has the following variations:


 * Dipentagonal prism - prism with dipentagons as bases, and 2 types of rectangles
 * Dipentagonal trapezoprism - isogonal with trapezoid sides
 * Decagonal frustum
 * dipentagonal frustum

Related polyhedra
A number of Johnson solids can be formed by attaching various configurations of pentagonal cupolas and pentagonal rotundas to the bases of the decagonal prism:


 * Elongated pentagonal cupola - Cupola attached to one base
 * Elongated pentagonal rotunda - Rotunda attached to one base
 * Elongated pentagonal orthobicupola - Cupolas in same orientation attached to both bases
 * Elongated pentagonal gyrobicupola - Cupolas rotated by 36º attached to bases
 * Elongated pentagonal orthocupolarotunda - Cpuola attached to one base, rotunda with same pentagon orientation attached to other base
 * Elongated pentagonal gyrocupolarotunda - Cupola attached to one base, rotunda with pentagon rotated by 36º attached to other base
 * Elongated pentagonal orthobirotunda - Rotundas in same orientation attached to both bases
 * Elongated pentagonal gyrobirotunda - Rotundas rotated by 36º attached to bases

The rhombisnub dodecahedron is a uniform polyhedron compound composed of 6 decagonal prisms.