# Decagonal prism

Decagonal prism
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymDip
Info
Coxeter diagramx x10o
SymmetryI2(10)×A1, order 40
ArmyDip
RegimentDip
Elements
Vertex figureIsosceles triangle, edge lengths 2, 2, (5+5)/2
Faces10 squares, 2 decagons
Edges10+20
Vertices20
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{7+2\sqrt5}}{2} ≈ 1.69353}$
Volume${\displaystyle 5\frac{\sqrt{5+2\sqrt5}}{2} ≈ 7.69421}$
Dihedral angles4–4: 144°
4–10: 90°
Height1
Central density1
Euler characteristic2
Number of pieces12
Level of complexity3
Related polytopes
DualDecagonal tegum
ConjugateDecagrammic prism
Properties
ConvexYes
OrientableYes
NatureTame

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:

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right).}$

## Representations

A decagonal prism has the following Coxeter diagrams:

## 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 ${\displaystyle \sqrt{a^2\frac{3+\sqrt5}{2}+\frac{b^2}{4}}}$ and its volume is given by ${\displaystyle 5\frac{\sqrt{5+2\sqrt5}}{2}a^2b}$.

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 ${\displaystyle \sqrt{\frac{5+\sqrt5}{2}}a}$ and side edges of lengths ${\displaystyle \sqrt{a^2+b^2}}$. In particular if the side edges are ${\displaystyle \frac{1+\sqrt5}{2}}$ times the length of the base edges this gives a uniform pentagonal antiprism.

## Variations

A decagonal prism has the following variations:

## 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:

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