Triangular prism (EntityTopic, 20)

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{{Shape|Triangular prism|''No image''|3|5, 9, 6|0|N/A|N/A|[[Line (object)|E]][[Triangle|T]]E|N/A|N/A|N/A|N/A|8|N/A|N/A|pure}}
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<[#ontology [kind topic] [cats Triangle Prism 3D Tapertope Uniform Polytope] [alt [[freebase:087c5y]] [[wikipedia:Triangular_prism]]]]>
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== Geometry ==
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{{STS Shape
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A '''triangular prism''' is a special case of a [[prism]] where the base is a [[triangle]].
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| dim=3
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| elements=2 [[triangle]]s, 3 [[square]]s, 9 [[digon]]s, 6 [[point]]s
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| sym=[[Pyroprismatic symmetry|D<sub>3h</sub>]]
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| genus=0
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| ssc=[G3x]
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| ssc2=+G3
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| extra={{STS Matrix|
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<i>2 2</i> <s>0 0</s>
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3 0 3 0
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1 1 1 1}}{{STS Tapertope
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| order=2, 1
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| notation=11<sup>1</sup>
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| index=10
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}}{{STS Polytope
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| flayout={{FLD|a3|i|a2|er}}
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| dual=[[Triangular bipyramid]]
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| bowers=Trip
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}}{{STS Uniform polytope
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| wythoff=<nowiki>2 3 | 2</nowiki>
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| schlaefli=t{2,3}
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| dynkin=x2x3o
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| conway=P3
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| vlayout=[[Triangle|3]]⋅[[Square|4]]<sup>2</sup>
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| vfigure=Isosceles [[triangle]], edges 1, √2, √2
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}}}}
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The triangular prism is the prism whose base is a triangle. If the sides are squares, it is a uniform polyhedron.
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=== Equations ===
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==Coordinates==
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The coordinates of a triangular prism of side 2 are:
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<blockquote>(±1, -√3/3, ±1)<br>(0, 2√3/3, ±1)</blockquote>
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== Equations ==
*Variables:
*Variables:
<blockquote>''l'' ⇒ length of edges of triangular prism</blockquote>
<blockquote>''l'' ⇒ length of edges of triangular prism</blockquote>
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*All points (''x'', ''y'', ''z'') that lie on the surface of a triangular prism< will satisfy the following equations:
 
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<blockquote>''Unknown''</blockquote>
 
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*All points (''x'', ''y'', ''z'') that lie on the edges of a triangular prism< will satisfy the following equations:
 
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<blockquote>''Unknown''</blockquote>
 
*The [[hypervolume]]s of a triangular prism are given by:
*The [[hypervolume]]s of a triangular prism are given by:
<blockquote>total edge length = 9''l''<br>
<blockquote>total edge length = 9''l''<br>
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surface area = ''l''<sup>2</sup>(3 + 2<sup>-1</sup>√3)<br>
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surface area = (3 + {{Over|√3|2}}) {{DotHV}}<br>
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volume = ''l''<sup>3</sup>4<sup>-1</sup>√3</blockquote>
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volume = {{Over|√3|4}} {{DotHV|3}}</blockquote>
*The [[planar]] [[cross-section]]s (''n'') of a triangular prism are:
*The [[planar]] [[cross-section]]s (''n'') of a triangular prism are:
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<blockquote>''Unknown''</blockquote>
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<blockquote>[!x,!y] ⇒ [[square]]
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<br clear="all"><br>
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[!z] ⇒ [[triangle]]</blockquote>
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<[#polytope [id 32]]>
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{{Polyhedra}}
{{Polyhedra}}
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{{Rotope Nav|7|8|9|(III)<br>Sphere|I'I<br>Triangular prism|<nowiki>I''</nowiki><br>Tetrahedron}}
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{{Tapertope Nav|9|10|11|[11]<sup>1</sup><br>Square pyramid|11<sup>1</sup><br>Triangular prism|1<sup>2</sup><br>Tetrahedron|hedra}}

Latest revision as of 14:45, 26 March 2017

The triangular prism is the prism whose base is a triangle. If the sides are squares, it is a uniform polyhedron.

Coordinates

The coordinates of a triangular prism of side 2 are:

(±1, -√3/3, ±1)
(0, 2√3/3, ±1)

Equations

  • Variables:
l ⇒ length of edges of triangular prism
total edge length = 9l
surface area = (3 + √3∕2) · l2
volume = √3∕4 · l3
[!x,!y] ⇒ square [!z] ⇒ triangle

Incidence matrix

Dual: triangular bipyramid

#TXIDVaEaEb4a3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 Eb 2 = digon ;
3 4a 422 = square ;
4 3a 303 = base of prism: triangle ;
5 C1a 63632 = triangular prism ;

Usage as facets


Notable Trishapes
Regular: tetrahedron • cube • octahedron • dodecahedron • icosahedron
Direct truncates: tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate
Mesotruncates: stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron
Snubs: snub staurohedron • snub rhodohedron
Curved: sphere • torus • cylinder • cone • frustum • crind


9. [11]1
Square pyramid 		10. 111
Triangular prism 		11. 12
Tetrahedron
List of tapertopes
Retrieved from "http://hi.gher.space/wiki/Triangular_prism"