View source for Elongated pentagonal cupola

{{Short description|20th Johnson solid}}
{{Infobox polyhedron
|image=elongated_pentagonal_cupola.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[Elongated square cupola|''J''{{sub|19}}]] – '''''J''{{sub|20}}''' – [[Elongated pentagonal rotunda|''J''{{sub|21}}]]}}
|faces=5 [[Triangle|triangle]]s<br>15 squares<br>1 [[Pentagon|pentagon]]<br>1 [[Decagon|decagon]]
|edges=45
|vertices=25
|symmetry={{math|''C''{{sub|5v}}}}
|vertex_config={{math|10(4{{sup|2}}.10)<br>10(3.4{{sup|3}})<br>5(3.4.5.4)}}
|dual=-
|properties=[[Convex polytope|convex]]
|net=Johnson solid 20 net.png
}}

In [[Geometry|geometry]], the '''elongated pentagonal cupola''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|20}}}}). As the name suggests, it can be constructed by elongating a [[Pentagonal cupola|pentagonal cupola]] ({{math|''J''{{sub|5}}}}) by attaching a [[Decagonal prism|decagonal prism]] to its base. The solid can also be seen as an [[Elongated pentagonal orthobicupola|elongated pentagonal orthobicupola]] ({{math|''J''{{sub|38}}}}) with its "lid" (another [[Pentagonal cupola|pentagonal cupola]]) removed.

{{Johnson solid}}

==Formulas==
The following formulas for the [[Physics:Volume|volume]] and [[Surface area|surface area]] can be used if all [[Face (geometry)|faces]] are [[Regular polygon|regular]], with edge length ''a'':<ref>[[Biography:Stephen Wolfram|Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=Elongated+pentagonal+cupola Elongated pentagonal cupola]" from [[Physics:Wolfram Alpha|Wolfram Alpha]]. Retrieved July 22, 2010.</ref>

:<math>V=\left(\frac{1}{6}\left(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)\right)a^3\approx10.0183...a^3</math>

:<math>A=\left(\frac{1}{4}\left(60+\sqrt{10\left(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}}\right)}\right)\right)a^2\approx26.5797...a^2</math>

=== Dual polyhedron ===

The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.
{| class=wikitable width=320
|- valign=top
!Dual elongated pentagonal cupola
!Net of dual
|- valign=top
|[[File:Dual elongated pentagonal cupola.png|160px]]
|[[File:Dual elongated pentagonal cupola net.png|160px]]
|}

==References==
{{Reflist}}

==External links==
* {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =ElongatedPentagonalCupola| title = Elongated pentagonal cupola}}

{{Johnson solids navigator}}
[[Category:Johnson solids]]

{{Sourceattribution|Elongated pentagonal cupola}}