Multiplicative group of integers modulo n: Difference between revisions

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{{Infobox polyhedron
|image=elongated_triangular_cupola.png
|type=[[Johnson solid|Johnson]]<br>[[gyroelongated square bipyramid|J<sub>17</sub>]] - '''J<sub>18</sub>''' - [[elongated square cupola|J<sub>19</sub>]]
|faces=1+3 [[triangle]]s<br>3.3 [[Square (geometry)|square]]s<br>1 [[hexagon]]
|edges=27
|vertices=15
|symmetry=''C''<sub>3v</sub>|
|vertex_config=6(4<sup>2</sup>.6)<br>3(3.4.3.4)<br>6(3.4<sup>3</sup>)
|dual=-
|properties=[[convex set|convex]]
|net=Johnson solid 18 net.png
}}


In [[geometry]], the '''elongated triangular cupola''' is one of the [[Johnson solid]]s  (''J''<sub>18</sub>). As the name suggests, it can be constructed by elongating a [[triangular cupola]] (''J''<sub>3</sub>) by attaching a [[hexagonal prism]] to its base.


{{Johnson solid}}
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==Formulae==
The following [[formula]]e for [[volume]] and [[surface area]] can be used if all [[face (geometry)|faces]] are [[regular polygon|regular]], with edge length ''a'':<ref>[[Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=Elongated+triangular+cupola Elongated triangular cupola]" from [[Wolfram Alpha]]. Retrieved July 22, 2010.</ref>
 
<math>V=(\frac{1}{6}(5\sqrt{2}+9\sqrt{3}))a^3\approx3.77659...a^3</math>
 
<math>A=(9+\frac{5\sqrt{3}}{2})a^2\approx13.3301...a^2</math>
 
=== Dual polyhedron ===
 
The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.
{| class=wikitable width=320
|- valign=top
!Dual elongated triangular cupola
!Net of dual
|- valign=top
|[[File:Dual elongated triangular cupola.png|160px]]
|[[File:Dual elongated triangular cupola net.png|160px]]
|}
 
==References==
{{Reflist}}
 
==External links==
* {{mathworld | urlname = JohnsonSolid | title = Johnson solid}}
* {{mathworld | urlname =ElongatedTriangularCupola| title =Elongated triangular cupola}}
 
[[Category:Johnson solids]]
 
 
{{Polyhedron-stub}}

Revision as of 02:10, 5 March 2014


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