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| {{Infobox polyhedron
| | The author is known as Irwin Wunder but it's not the most masucline name out there. Years in the past we moved to North Dakota. Managing individuals is what I do and the wage has been really satisfying. It's not a common thing but what she likes doing is foundation leaping and now she is attempting to earn money with it.<br><br>My site; diet meal delivery ([http://bgo.url.ph/weightlossfoodprograms67803 navigate to this site]) |
| |image=pentagonal_cupola.png
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| |type=[[Johnson solid|Johnson]]<br>[[square cupola|J<sub>4</sub>]] - '''J<sub>5</sub>''' - [[pentagonal rotunda|J<sub>6</sub>]]
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| |faces=5 [[triangle]]s<br>5 [[Square (geometry)|square]]s<br>1 [[pentagon]]<br>1 [[decagon]]
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| |edges=25
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| |vertices=15
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| |symmetry=''C''<sub>5v</sub>, [5], (*55)
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| |rotation_group=''C''<sub>5</sub>, [5]<sup>+</sup>, (55)
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| |vertex_config=10(3.4.10)<br>5(3.4.5.4)
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| |dual=-
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| |properties=[[convex set|convex]]
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| |net=Pentagonal Cupola.PNG
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| }}
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| In [[geometry]], the '''pentagonal [[cupola (geometry)|cupola]]''' is one of the [[Johnson solid]]s (''J''<sub>5</sub>). It can be obtained as a slice of the [[rhombicosidodecahedron]]. The pentagonal cupola consists of 5 [[equilateral triangle]]s, 5 [[Square (geometry)|square]]s, 1 [[pentagon]], and 1 [[decagon]].
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| {{Johnson solid}}
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| ==Formulae==
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| The following [[formula]]e for [[volume]], [[surface area]] and [[circumscribed sphere|circumradius]] 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=Pentagonal+cupola Pentagonal cupola]" from [[Wolfram Alpha]]. Retrieved July 21, 2010.</ref>
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| <math>V=\left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3\approx2.32405...a^3</math>
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| <math>A=\left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5(145+62\sqrt{5})}\right)\right)a^2=\left(\frac{1}{4}\left(20+\sqrt{10\left(80+31\sqrt{5}+\sqrt{15(145+62\sqrt{5})}\right)}\right)\right)a^2\approx16.5797...a^2</math>
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| <math>C=\left(\frac{1}{2}\sqrt{11+4\sqrt{5}}\right)a\approx2.23295...a</math>
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| == Related polyhedra==
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| === Dual polyhedron ===
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| The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces:
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| {| class=wikitable width=320
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| |- valign=top
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| !Dual pentagonal cupola
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| !Net of dual
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| |- valign=top
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| |[[File:Dual pentagonal cupola.png|160px]]
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| |[[File:Dual pentagonal cupola net.png|160px]]
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| |}
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| {{Cupolae}}
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{mathworld | urlname = JohnsonSolid | title = Johnson solid}}
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| * {{mathworld | urlname =PentagonalCupola| title =Pentagonal cupola}}
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Johnson solids]]
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| {{Polyhedron-stub}}
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The author is known as Irwin Wunder but it's not the most masucline name out there. Years in the past we moved to North Dakota. Managing individuals is what I do and the wage has been really satisfying. It's not a common thing but what she likes doing is foundation leaping and now she is attempting to earn money with it.
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