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{{Prism polyhedra db|Prism polyhedron stat table|P5}} | |||
In [[geometry]], the '''pentagonal prism''' is a [[Prism (geometry)|prism]] with a [[pentagon]]al base. It is a type of [[heptahedron]] with 7 [[Face (geometry)|faces]], 15 [[Edge (geometry)|edges]], and 10 [[vertex (geometry)|vertices]]. | |||
== As a semiregular (or uniform) polyhedron == | |||
If faces are all regular, the pentagonal prism is a [[semiregular polyhedron]], more generally, a [[uniform polyhedron]], and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a '''[[truncation (geometry)|truncated]] [[hosohedron|pentagonal hosohedron]]''', represented by [[Schläfli symbol]] t{2,5}. Alternately it can be seen as the [[Cartesian product]] of a regular pentagon and a [[line segment]], and represented by the product {5}x{}. The [[dual polyhedron|dual]] of a pentagonal prism is a [[pentagonal bipyramid]]. | |||
The [[symmetry group]] of a right pentagonal prism is [[dihedral group|''D<sub>5h</sub>'']] of order 20. The [[Point groups in three dimensions#Rotation groups|rotation group]] is ''D<sub>5</sub>'' of order 10. | |||
==Volume== | |||
The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism with edges ''h'' the formula is | |||
:<math>\frac{h^3}{4}\sqrt{5(5 + 2\sqrt{5})}</math> | |||
== Use == | |||
Nonuniform pentagonal prisms called [[Pentaprism]]s are also used in optics to rotate an image through a [[right angle]] without changing its [[chirality]]. | |||
=== In polychora === | |||
It exists as cells of four nonprismatic [[uniform polychoron|polychora]] in 4 dimensions: | |||
{| class=wikitable width=400 | |||
|- align=center | |||
|[[cantellated 600-cell]]<BR>{{CDD|node|5|node_1|3|node|3|node_1}} | |||
|[[cantitruncated 600-cell]]<BR>{{CDD|node|5|node_1|3|node_1|3|node_1}} | |||
|[[runcinated 600-cell]]<BR>{{CDD|node_1|5|node|3|node|3|node_1}} | |||
|[[runcitruncated 600-cell]]<BR>{{CDD|node_1|5|node|3|node_1|3|node_1}} | |||
|- align=center | |||
|[[File:600-cell t02 H3.svg|100px]] | |||
|[[File:120-cell t123 H3.png|100px]] | |||
|[[File:120-cell_t03_H3.png|100px]] | |||
|[[File:120-cell_t023_H3.png|100px]] | |||
|} | |||
== Related polyhedra == | |||
{{UniformPrisms}} | |||
==External links == | |||
* {{mathworld | urlname = PentagonalPrism | title = Pentagonal prism}} | |||
*[http://polyhedra.org/poly/show/23/pentagonal_prism Pentagonal Prism Polyhedron Model] -- works in your web browser | |||
[[Category:Prismatoid polyhedra]] | |||
{{Polyhedron-stub}} | |||
Revision as of 15:04, 3 February 2014
Template:Prism polyhedra db In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.
As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}x{}. The dual of a pentagonal prism is a pentagonal bipyramid.
The symmetry group of a right pentagonal prism is D5h of order 20. The rotation group is D5 of order 10.
Volume
The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism with edges h the formula is
Use
Nonuniform pentagonal prisms called Pentaprisms are also used in optics to rotate an image through a right angle without changing its chirality.
In polychora
It exists as cells of four nonprismatic polychora in 4 dimensions:
| cantellated 600-cell Template:CDD |
cantitruncated 600-cell Template:CDD |
runcinated 600-cell Template:CDD |
runcitruncated 600-cell Template:CDD |
|
|
|
|
Related polyhedra
External links
- 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
Here is my web site - cottagehillchurch.com - Pentagonal Prism Polyhedron Model -- works in your web browser