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| {{For|the optical prism|Triangular prism (optics)}}
| | Hello and welcome. My title is Irwin and I totally dig that title. Body building is what my family members and I enjoy. Her husband and her live in Puerto Rico but she will have to move 1 working day or an additional. He used to be unemployed but now he is a meter reader.<br><br>Have a look at my homepage :: std testing at home ([http://203.250.78.160/zbxe/?document_srl=817810&mid=gallery visit the website]) |
| {{Unreferenced|date=February 2011}}
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| {{Prism polyhedra db|Prism polyhedron stat table|P3}}
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| In [[geometry]], a '''triangular prism''' is a three-sided [[Prism (geometry)|prism]]; it is a [[polyhedron]] made of a [[triangle|triangular]] base, a [[Translation (geometry)|translated]] copy, and 3 faces joining corresponding sides.
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| Equivalently, it is a [[pentahedron]] of which two faces are parallel, while the [[surface normal]]s of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are [[parallelogram]]s. All cross-sections parallel to the base faces are the same triangle.
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| == As a semiregular (or uniform) polyhedron ==
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| A right triangular prism is [[semiregular polyhedron|semiregular]] or, more generally, a [[uniform polyhedron]] if the base faces are equilateral [[triangle]]s, and the other three faces are [[square (geometry)|squares]]. It can be seen as a '''[[truncation (geometry)|truncated]] [[hosohedron|trigonal hosohedron]]''', represented by [[Schläfli symbol]] t{2,3}. Alternately it can be seen as the [[Cartesian product]] of a triangle and a [[line segment]], and represented by the product {3}x{}. The [[dual polyhedron|dual]] of a triangular prism is a [[triangular bipyramid]].
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| The [[symmetry group]] of a right 3-sided prism with triangular base is [[dihedral group|''D<sub>3h</sub>'']] of order 12. The [[Point groups in three dimensions#Rotation groups|rotation group]] is ''D<sub>3</sub>'' of order 6. The symmetry group does not contain [[Inversion in a point|inversion]].
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| == Volume ==
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| The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to [[Triangle#Computing the area of a triangle|compute the area of the triangle]] and multiply this by the length of the prism:
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| <math>V = \frac{1}{2} bhl</math>
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| where b is the triangle base length, h is the triangle height, and l is the length between the triangles.
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| == Related polyhedra and tilings ==
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| {{UniformPrisms}}
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| {{Cupolae}}
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| This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2n.2n), and [n,3] [[Coxeter group]] symmetry.
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| {{Truncated figure1 table}}
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| This polyhedron is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].
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| This polyhedron is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].
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| {{Expanded table}}
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| === Compounds ===
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| There are 4 uniform compounds of triangular prisms:
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| :[[Compound of four triangular prisms]], [[compound of eight triangular prisms]], [[compound of ten triangular prisms]], [[compound of twenty triangular prisms]].
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| === Honeycombs ===
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| There are 9 uniform honeycombs that include triangular prism cells:
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| :[[Gyroelongated alternated cubic honeycomb]], [[elongated alternated cubic honeycomb]], [[gyrated triangular prismatic honeycomb]], [[snub square prismatic honeycomb]], [[triangular prismatic honeycomb]], [[triangular-hexagonal prismatic honeycomb]], [[truncated hexagonal prismatic honeycomb]], [[rhombitriangular-hexagonal prismatic honeycomb]], [[snub triangular-hexagonal prismatic honeycomb]], [[elongated triangular prismatic honeycomb]]
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| === Related polytopes ===
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| The triangular prism is first in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es ([[equilateral triangle]]s and [[square]]s in the case of the triangular prism). In [[Coxeter]]'s notation the triangular prism is given the symbol −1<sub>21</sup>.
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| {{Gosset_semiregular_polytopes}}
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| === Four dimensional space===
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| The triangular prism exists as cells of a number of four-dimensional [[uniform polychoron|uniform polychora]], including:
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| {| class=wikitable width=640
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| |- align=center
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| |[[tetrahedral prism]]<BR>{{CDD|node_1|3|node|3|node|2|node_1}}
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| |[[octahedral prism]]<BR>{{CDD|node_1|3|node|4|node|2|node_1}}
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| |[[cuboctahedral prism]]<BR>{{CDD|node|3|node_1|4|node|2|node_1}}
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| |[[icosahedral prism]]<BR>{{CDD|node_1|3|node|5|node|2|node_1}}
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| |[[icosidodecahedral prism]]<BR>{{CDD|node|3|node_1|5|node|2|node_1}}
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| |[[Truncated dodecahedral prism]]<BR>{{CDD|node|3|node_1|5|node_1|2|node_1}}
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| |- align=center
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| |[[File:tetrahedral prism.png|80px]]
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| |[[File:octahedral prism.png|80px]]
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| |[[File:cuboctahedral prism.png|80px]]
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| |[[File:icosahedral prism.png|80px]]
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| |[[File:icosidodecahedral prism.png|80px]]
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| |[[File:Truncated dodecahedral prism.png|80px]]
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| |- align=center
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| |[[Rhombicosidodecahedral prism|Rhombi-cosidodecahedral prism]]<BR>{{CDD|node_1|3|node|5|node_1|2|node_1}}
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| |[[Rhombicuboctahedral prism|Rhombi-cuboctahedral prism]]<BR>{{CDD|node_1|3|node|4|node_1|2|node_1}}
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| |[[Truncated cubic prism]]<BR>{{CDD|node|3|node_1|4|node_1|2|node_1}}
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| |[[Snub dodecahedral prism]]<BR>{{CDD|node_h|5|node_h|3|node_h|2|node_1}}
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| |[[Uniform antiprismatic prism|n-gonal antiprismatic prism]]<BR>{{CDD|node_h|n|node_h|2x|node_h|2|node_1}}
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| |- align=center
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| |[[File:Rhombicosidodecahedral prism.png|80px]]
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| |[[File:Rhombicuboctahedral prism.png|80px]]
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| |[[File:Truncated cubic prism.png|80px]]
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| |[[File:Snub dodecahedral prism.png|80px]]
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| |[[File:Square antiprismatic prism.png|80px]]
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| |- align=center
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| |[[Cantellated 5-cell]]<BR>{{CDD|node_1|3|node|3|node_1|3|node}}
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| |[[Cantitruncated 5-cell]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node}}
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| |[[Runcinated 5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
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| |[[Runcitruncated 5-cell]]<BR>{{CDD|node_1|3|node_1|3|node|3|node_1}}
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| |[[Cantellated tesseract]]<BR>{{CDD|node_1|4|node|3|node_1|3|node}}
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| |[[Cantitruncated tesseract]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node}}
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| |[[Runcinated tesseract]]<BR>{{CDD|node_1|4|node|3|node|3|node_1}}
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| |[[Runcitruncated tesseract]]<BR>{{CDD|node_1|4|node_1|3|node|3|node_1}}
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| |- align=center
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| |[[File:4-simplex t02.svg|80px]]
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| |[[File:4-simplex t012.svg|80px]]
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| |[[File:4-simplex t03.svg|80px]]
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| |[[File:4-simplex t013.svg|80px]]
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| |[[File:4-cube t02.svg|80px]]
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| |[[File:4-cube t012.svg|80px]]
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| |[[File:4-cube t03.svg|80px]]
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| |[[File:4-cube t013.svg|80px]]
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| |- align=center
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| |[[Cantellated 24-cell]]<BR>{{CDD|node_1|3|node|4|node_1|3|node}}
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| |[[Cantitruncated 24-cell]]<BR>{{CDD|node_1|3|node_1|4|node_1|3|node}}
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| |[[Runcinated 24-cell]]<BR>{{CDD|node_1|3|node|4|node|3|node_1}}
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| |[[Runcitruncated 24-cell]]<BR>{{CDD|node_1|3|node_1|4|node|3|node_1}}
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| |[[Cantellated 120-cell]]<BR>{{CDD|node_1|5|node|3|node_1|3|node}}
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| |[[Cantitruncated 120-cell]]<BR>{{CDD|node_1|5|node_1|3|node_1|3|node}}
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| |[[Runcinated 120-cell]]<BR>{{CDD|node_1|5|node|3|node|3|node_1}}
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| |[[Runcitruncated 120-cell]]<BR>{{CDD|node_1|5|node_1|3|node|3|node_1}}
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| |- align=center
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| |[[File:24-cell t02 F4.svg|80px]]
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| |[[File:24-cell t012 F4.svg|80px]]
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| |[[File:24-cell t03 F4.svg|80px]]
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| |[[File:24-cell t013 F4.svg|80px]]
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| |[[File:120-cell t02 H3.png|80px]]
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| |[[File:120-cell t012 H3.png|80px]]
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| |[[File:120-cell t03 H3.png|80px]]
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| |[[File:120-cell t013 H3.png|80px]]
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| |}
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| == See also==
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| * [[Wedge (geometry)]]
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| ==External links==
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| * {{mathworld | urlname = TriangularPrism | title = Triangular prism}}
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| *[http://polyhedra.org/poly/show/22/triangular_prism Interactive Polyhedron: Triangular Prism]
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| * Whole site dedicated to triangular prisms. Good resource for high school students to learn about how to find the [http://www.triangular-prism.com surface area and volume of a triangular prism] and how to draw its net.
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Space-filling polyhedra]]
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Hello and welcome. My title is Irwin and I totally dig that title. Body building is what my family members and I enjoy. Her husband and her live in Puerto Rico but she will have to move 1 working day or an additional. He used to be unemployed but now he is a meter reader.
Have a look at my homepage :: std testing at home (visit the website)