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| {{Even polygon db|Even polygon stat table|p10}}
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| [[File:Gonbad-e Qabus.JPG|thumb|[[Gonbad-e Qabus (tower)|Gonbad-e Qabus]], the tallest pure brick tower in the world, is built on a decagonal plan.]]
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| In [[geometry]], a '''decagon''' is any [[polygon]] with ten sides and ten [[angle]]s. A [[regular polygon|regular]] decagon has all sides of equal length and each internal angle equal to 144°. Its [[Schläfli symbol]] is {10}.
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| ==Regular decagon==
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| The [[area]] of a regular decagon is: (with ''t'' = edge length) | |
| :<math>A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2.</math>
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| An alternative formula is <math>\scriptstyle A\,=\,2.5dt</math> where ''d'' is the distance between parallel sides, or the height when the decagon stands on one side as base.<br>
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| By simple trigonometry <math>\scriptstyle d\,=\,2t(\cos{54^\circ}\,+\,\cos{18^\circ})</math>.
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| ==Sides==
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| The side of a regular decagon inscribed in a unit circle is <math>\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}</math>, where ''ϕ'' is the [[golden ratio]], <math>\tfrac{1+\sqrt{5}}{2}</math>.
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| ===Construction===
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| A regular decagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:
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| [[File:Regular Decagon Inscribed in a Circle.gif|Construction of a regular decagon]]
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| An alternative (but similar) method is as follows:
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| #Construct a pentagon in a circle by one of the methods shown in [[Pentagon#Construction_of_a_regular_pentagon|constructing a pentagon]].
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| #Extend a line from each vertex of the pentagon through the center of the [[circle]] to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.
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| #The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.
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| ==Related figures==
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| There is one regular [[star polygon]], the [[decagram (geometry)|decagram]] {10/3}, using the same points, but connecting every third points. There are also two compounds: {10/4} is reduced to 2{5/2} as two [[pentagram]]s, and {10/2} is reduced to 2{5} as two [[pentagon]]s.
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| {| class=wikitable width=360
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| |- align=center
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| |[[File:Truncated pentagon.png|120px]]<BR>A [[truncation (geometry)|truncated]] regular pentagon
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| |[[File:Decagram_10_3.png|120px]]<br>{10/3}<BR>[[Decagram (geometry)|Decagram]]
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| |[[Image:Decagram 10 2.png|120px]]<br>{10/2} or 2{5}
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| |[[Image:Decagram 10 4.png|120px]]<br>{10/4} or 2{5/2}
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| |}
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| ===Petrie polygons===
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| The regular decagon is the [[Petrie polygon]] for many higher dimensional polytopes, shown in these skew [[orthogonal projection]]s in various [[Coxeter plane]]s:
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| {| class=wikitable width=450
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| |- align=center valign=top
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| !valign=center|A<sub>9</sub>
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| |[[File:9-simplex_t0.svg|100px]]<br>[[9-simplex]]
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| |[[File:9-simplex_t1.svg|100px]]<br>[[Rectified 9-simplex]]
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| |[[File:9-simplex_t2.svg|100px]]<br>[[Birectified 9-simplex]]
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| |[[File:9-simplex_t3.svg|100px]]<br>[[Trirectified 9-simplex]]
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| |[[File:9-simplex_t4.svg|100px]]<br>[[Quadrirectified 9-simplex]]
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| |- align=center valign=top
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| !valign=center|BC<sub>5</sub>
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| |[[File:5-cube_t4.svg|100px]]<br>[[5-orthoplex]]
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| |[[File:5-cube_t3.svg|100px]]<br>[[Rectified 5-orthoplex]]
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| |[[File:5-cube_t2.svg|100px]]<br>[[Birectified 5-cube]]
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| |[[File:5-cube_t1.svg|100px]]<br>[[Rectified 5-cube]]
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| |[[File:5-cube_t0.svg|100px]]<br>[[5-cube]]
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| |- align=center valign=top
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| !valign=center|D<sub>6</sub>
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| |[[File:6-cube_t5_B5.svg|100px]]<br>[[5-orthoplex|t<sub>1</sub>(4<sub>31</sub>)]]
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| |[[File:6-cube_t4_B5.svg|100px]]<br>[[Rectified 5-orthoplex|t<sub>3</sub>(1<sub>31</sub>)]]
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| |[[File:6-cube_t3_B5.svg|100px]]<br>[[Birectified 5-orthoplex|t<sub>2</sub>(1<sub>31</sub>)]]
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| |[[File:6-demicube_t1_D6.svg|100px]]<br>[[Rectified 6-demicube|t<sub>1</sub>(1<sub>31</sub>)]]
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| |[[File:6-demicube_t0_D6.svg|100px]]<br>[[6-demicube]]<br>(1<sub>31</sub>)
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| |- align=center valign=top
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| !valign=center|H<sub>3</sub>
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| |[[File:Dodecahedron petrie.png|100px]]<br>[[Dodecahedron]]
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| |[[File:Icosahedron petrie.png|100px]]<br>[[Icosahedron]]
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| |[[File:Dodecahedron t1 H3.png|100px]]<br>[[Icosidodecahedron]]
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| |}
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| ==See also==
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| *[[decagonal number]]
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| *[[Gambrel]]
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| *[[Golden ratio]]
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| ==External links==
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| *{{MathWorld |urlname=Decagon |title=Decagon}}
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| *[http://www.mathopenref.com/decagon.html Definition and properties of a decagon] With interactive animation
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| {{Polygons}}
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| [[Category:Polygons]]
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