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| {{Infobox polyhedron
| | I woke up last week and realised - I've been single for a while at the moment and after much intimidation from pals I now find myself registered for on line dating. They [http://Www.Bing.com/search?q=promised&form=MSNNWS&mkt=en-us&pq=promised promised] me that there are plenty of ordinary, sweet and enjoyable individuals to meet, therefore here goes the pitch!<br>My fam and pals are wonderful and spending some time with them at tavern gigs or dishes luke bryan 2014 - [http://lukebryantickets.neodga.com see this website], is obviously critical. As I see that one may never own a decent conversation with the sound I have never been in to dance clubs. Additionally, I have 2 quite cute and unquestionably cheeky canines who are almost always enthusiastic to meet fresh folks.<br>I try and stay [http://lukebryantickets.lazintechnologies.com luke bryan Discount Tickets] as physically healthy as possible being at the gymnasium several times a week. I enjoy my athletics and endeavor to perform or view while many a possible. Being winter I will regularly at Hawthorn matches. Notice: I have experienced the carnage of fumbling matches at stocktake sales, If you really considered shopping a sport I really do not brain.<br><br>My webpage [http://www.hotelsedinburgh.org luke bryan concert today] |
| |image=Pentagonal pyramid.png
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| |type=[[Johnson solid|Johnson]]<br>[[square pyramid|J<sub>1</sub>]] - '''J<sub>2</sub>''' - [[triangular cupola|J<sub>3</sub>]]
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| |faces=5 [[triangle]]s<br>1 [[pentagon]]
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| |edges=10
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| |vertices=6
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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=5(3<sup>2</sup>.5)<br>(3<sup>5</sup>)
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| |dual=self
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| |properties=[[convex set|convex]]
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| |net=pentagonal pyramid flat.svg
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| }}
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| In [[geometry]], a '''pentagonal pyramid''' is a [[Pyramid (geometry)| pyramid]] with a [[pentagon]]al base upon which are erected five [[triangle|triangular]] faces that meet at a point (the vertex). Like any [[pyramid]], it is self-[[dual polyhedron|dual]].
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| The ''regular'' pentagonal pyramid has a base that is a regular pentagon and lateral faces that are [[equilateral triangle]]s. It is one of the [[Johnson solid]]s (''J''<sub>2</sub>). Its height ''H'', from the midpoint of the pentagonal face to the apex, (as a function of ''a'', where ''a'' is the side length), can be computed as:
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| :<math>H = \sqrt{{\frac{5-\sqrt{5}}{10}}}\,a \approx 0.5257\,a.</math><!--unchecked-->
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| Its surface area, ''A'', can be computed as the area of pentagonal base plus five times the area of one triangle:
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| :<math>A = \left( \frac{\sqrt{25 + 10 \sqrt{5}}}{4} + 5\frac{\sqrt{3}}{4} \right) a^2 \approx 3.8855\,a^2.</math><!--checked-->
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| Its volume when an edge length is known can be figured out with this formula:
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| :<math>V = \frac{5 + \sqrt{5}}{24}\,a^3 \approx 0.3015\,a^3.</math><!--unchecked--> | |
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| It can be seen as the "lid" of an [[icosahedron]]; the rest of the icosahedron forms a [[gyroelongated pentagonal pyramid]], ''J''<sub>11</sub>, one of the 92 Johnson solids named and described by [[Norman Johnson (mathematician)|Norman Johnson]] in 1966.
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| More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height.
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| == Related polyhedra ==
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| {{Pyramids}}
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| {| class=wikitable width=320
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| |[[File:Pentagonal frustum.svg|160px]]<br>Pentagonal [[frustum]] is a pentagonal pyramid with its apex truncated
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| |[[File:Icosahedron.png|160px]]<br>The vertex of an [[icosahedron]] is a pentagonal pyramid
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| |}
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| === Dual polyhedron ===
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| The pentagonal pyramid is topologically a [[self-dual polyhedron]]. The dual edge lengths are different due to the [[polar reciprocation]].
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| {| class=wikitable width=320
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| |- valign=top
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| !Dual pentagonal pyramid
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| !Net of dual
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| |- valign=top
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| |[[File:Dual pentagonal pyramid.png|162px]]
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| |[[File:Dual pentagonal pyramid net.png|160px]]
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| |}
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| ==External links==
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| * {{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname = PentagonalPyramid | title = Pentagonal pyramid}}
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra ( [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_pyramid_(J2).wrl model])
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| [[Category:Pyramids and bipyramids]]
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| [[Category:Self-dual polyhedra]]
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Johnson solids]]
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| {{Polyhedron navigator}}
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