View source for Decagon

{{Short description|Shape with ten sides}}
{{Regular polygon db|Regular polygon stat table|p10}}
In [[Geometry|geometry]], a '''decagon''' (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided [[Polygon|polygon]] or 10-gon.<ref name="sidebotham">{{citation|title=The A to Z of Mathematics: A Basic Guide|first=Thomas H.|last=Sidebotham|publisher=John Wiley & Sons|year=2003|isbn=9780471461630|page=146|url=https://books.google.com/books?id=VsAZa5PWLz8C&pg=PA146}}.</ref> The total sum of the [[Internal and external angles|interior angles]] of a [[Simple polygon|simple]] decagon is 1440°.
==Regular decagon==
A ''[[Regular polygon|regular]] decagon'' has all sides of equal length and each internal angle will always be equal to 144°.<ref name="sidebotham"/> Its [[Schläfli symbol]] is {10} <ref>{{citation|title=Polyhedron Models|first=Magnus J.|last=Wenninger|publisher=Cambridge University Press|year=1974|page=9|isbn=9780521098595|url=https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9}}.</ref> and can also be constructed as a [[Truncation (geometry)|truncated]] [[Pentagon|pentagon]], t{5}, a quasiregular decagon alternating two types of edges.

=== Side length ===
[[File:01-Zehneck-Seitenlänge.svg|300px|right]]
The picture shows a regular decagon with side length <math>a</math> and radius <math>R</math> of the [[Circumscribed circle|circumscribed circle]].

* The triangle <math>E_{10}E_1M</math> has two equally long legs with length <math>R</math> and a base with length <math>a</math>
* The circle around <math>E_1</math> with radius <math>a</math> intersects <math>]M\,E_{10}[</math> in a point <math>P</math> (not designated in the picture).
* Now the triangle <math>{E_{10}E_1P}\;</math> is an [[Isosceles triangle|isosceles triangle]] with vertex <math>E_1</math> and with base angles <math>m\angle E_1 E_{10} P = m\angle E_{10} P E_1 = 72^\circ \;</math>.
* Therefore <math>m\angle P E_1 E_{10} = 180^\circ -2\cdot 72^\circ = 36^\circ \;</math>. So <math>\; m\angle M E_1 P = 72^\circ- 36^\circ = 36^\circ\;</math> and hence <math>\; E_1 M P\;</math> is also an isosceles triangle with vertex <math>P</math>. The length of its legs is <math>a</math>, so the length of <math>[P\,E_{10}]</math> is <math>R-a</math>.
* The isosceles triangles <math>E_{10} E_1 M\;</math> and <math>P E_{10} E_1\;</math> have equal angles of 36° at the vertex, and so they are [[Similarity (geometry)|similar]], hence: <math>\;\frac{a}{R}=\frac{R-a}{a}</math>
* Multiplication with the denominators <math>R,a >0</math> leads to the quadratic equation: <math>\;a^2=R^2-aR\;</math>
* This equation for the side length <math>a\,</math> has one positive solution: <math>\;a=\frac{R}{2}(-1+\sqrt{5})</math>
So the regular decagon can be constructed with ''ruler and compass''.

;Further conclusions:
<math>\;R=\frac{2a}{\sqrt{5}-1}=\frac{a}{2}(\sqrt{5}+1)\;</math> and the base height of <math>\Delta\,E_{10} E_1 M\,</math> (i.e. the length of <math>[M\,D]</math>) is <math>h = \sqrt{R^2-(a/2)^2}=\frac{a}{2}\sqrt{5+2\sqrt{5}}\;</math> and the triangle has the area: <math>A_\Delta=\frac{a}{2}\cdot h = \frac{a^2}{4}\sqrt{5+2\sqrt{5}}</math>.

===Area===
The [[Area|area]] of a regular decagon of side length ''a'' is given by:<ref>{{citation|title=The elements of plane and spherical trigonometry|publisher=Society for Promoting Christian Knowledge|year=1850|page=59|url=https://books.google.com/books?id=AW7qzTr_f1sC&pg=PA59}}. Note that this source uses ''a'' as the edge length and gives the argument of the cotangent as an angle in degrees rather than in radians.</ref>
:<math> A = \frac{5}{2} a^2\cot\left(\frac{\pi}{10} \right) = 
                     \frac{5}{2} a^2\sqrt{5+2\sqrt{5}}
                \simeq 7.694208843\,a^2
 </math>

In terms of the [[Apothem|apothem]] ''r'' (see also [[Inscribed figure|inscribed figure]]), the area is:
:<math>A = 10 \tan\left(\frac{\pi}{10}\right) r^2 = 
                     2r^2\sqrt{5\left(5-2\sqrt5\right)}
                 \simeq 3.249196962\,r^2
</math>

In terms of the circumradius ''R'', the area is:
:<math>
A = 5 \sin\left(\frac{\pi}{5}\right) R^2
= \frac{5}{2}R^2\sqrt{\frac{5-\sqrt{5}}{2}}
\simeq 2.938926261\,R^2
</math>

An alternative formula is <math>A=2.5da</math> where ''d'' is the distance between parallel sides, or the height when the decagon stands on one side as base, or the [[Diameter|diameter]] of the decagon's [[Inscribed figure|inscribed circle]].
By simple [[Trigonometry|trigonometry]],

:<math>d=2a\left(\cos\tfrac{3\pi}{10}+\cos\tfrac{\pi}{10}\right),</math>

and it can be written [[Algebraic expression|algebraically]] as

:<math>d=a\sqrt{5+2\sqrt{5}}.</math>

===Sides===
A regular decagon has 10 sides and is equilateral. It has 35 diagonals

===Construction===
As 10 = 2 × 5, a [[Power of two|power of two]] times a Fermat prime, it follows that a regular decagon is [[Constructible polygon|constructible]] using compass and straightedge, or by an edge-[[Bisection|bisection]] of a regular [[Pentagon|pentagon]].<ref name="ludlow">{{citation|title=Geometric Construction of the Regular Decagon and Pentagon Inscribed in a Circle|first=Henry H.|last=Ludlow|publisher=The Open Court Publishing Co.|year=1904|url=https://books.google.com/books?id=vLMlw7uL8kgC}}.</ref>
{{multiple image
| align = left
| image1 = Regular Decagon Inscribed in a Circle.gif
| width1 = 260
| alt1 = 
| caption1 = Construction of decagon
| image2 = Regular Pentagon Inscribed in a Circle.gif
| width2 = 260
| alt2 = 
| caption2 = Construction of pentagon
| footer = 
}}
{{clear}}

An alternative (but similar) method is as follows:
#Construct a pentagon in a circle by one of the methods shown in [[Pentagon#Construction of a regular pentagon|constructing a pentagon]].
#Extend a line from each vertex of the pentagon through the center of the [[Circle|circle]] to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.  In other words,  [[Image (mathematics)#Image of a subset|the image]] of a regular pentagon under a point reflection with respect of [[Regular polygon#Symmetry|its center]] is a [[Concentric objects|concentric]] ''[[Congruence (geometry)|congruent]]'' pentagon,  and the two pentagons have in total the vertices of a concentric ''regular decagon''.
#The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.

===Nonconvex regular decagon===

[[File:Golden tiling with rotational symmetry.svg|thumb|left|This [[Tessellation|tiling]] by [[Golden triangle (mathematics)|golden triangles]], a regular [[Pentagon|pentagon]], contains a stellation of regular decagon, the [[Schläfli symbol|Schäfli&nbsp;symbol]] of which is {10/3}.]]
The length [[Ratio|ratio]] of two inequal edges of a golden triangle is the [[Golden ratio|golden ratio]], denoted by <math>\Phi</math>, or its [[Multiplicative inverse|multiplicative&nbsp;inverse]]:
<math display="block"> \Phi - 1 = \frac{1}{\Phi} = 2\,\cos 72\,^\circ = \frac{1}{\,2\,\cos 36\,^\circ} = \frac{\,\sqrt{5} - 1\,}{2} \text{.}</math>
So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this [[Star polygon|star&nbsp;polygon]].
{{Clear}}

== The golden ratio in decagon ==
Both in the construction with given circumcircle<ref name="Henry Green">{{citation|title=Euclid's Plane Geometry, Books III–VI, Practically Applied, or Gradations in Euclid, Part II|first=Henry|last=Green|publisher=London: Simpkin, Marshall,& CO.| year=1861|page=116|url=https://books.google.com/books?id=DjQDAAAAQAAJ&q=II.+Practically+decagon+Euklid+p.+116&pg=PA116}}. Retrieved 10 February 2016.</ref> as well as with given side length is the [[Golden ratio#Geometry|golden ratio dividing a line segment by exterior division]] the
determining construction element.

* In the construction with given circumcircle the circular arc around G with radius {{Overline|GE<sub>3</sub>}} produces the segment {{Overline|AH}}, whose division corresponds to the golden ratio.
:<math>\frac{\overline{AM}}{\overline{MH}} = \frac{\overline{AH}}{\overline{AM}} = \frac{1+ \sqrt{5}}{2} = \Phi \approx 1.618 \text{.}</math>

* In the construction with given side length<ref name="Jürgen Köller">{{citation|title=Regelmäßiges Zehneck, → 3. Section "Formeln, Ist die Seite a gegeben ..." |first=Jürgen|last=Köller| year=2005|url=http://www.mathematische-basteleien.de/zehneck.htm|language=de}}. Retrieved 10 February 2016.</ref> the circular arc around D with radius {{Overline|DA}} produces the segment {{Overline|E<sub>10</sub>F}}, whose division corresponds to the [[Golden ratio#Calculation|golden ratio]].
:<math>\frac{\overline{E_1 E_{10}}}{\overline{E_1 F}} = \frac{\overline{E_{10} F}}{\overline{E_1 E_{10}}} = \frac{R}{a} = \frac{1+ \sqrt{5}}{2} =\Phi \approx 1.618 \text{.}</math>
{{multiple image
| align = left
| image1 = 01-Zehneck-Animation.gif
| width1 = 287
| alt1 = 
| caption1 = Decagon with given circumcircle,<ref name="Henry Green" /> animation
| image2 = 01-Zehneck-Seitenlänge-Animation.gif
| width2 = 300
| alt2 = 
| caption2 = Decagon with a given side length,<ref name="Jürgen Köller" /> animation
| footer = 
}}
{{clear}}

== Symmetry==
[[File:Symmetries_of_decagon.png|thumb|320px|Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center.]]

The ''regular decagon'' has Dih<sub>10</sub> symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih<sub>5</sub>, Dih<sub>2</sub>, and Dih<sub>1</sub>, and 4 [[Cyclic group|cyclic group]] symmetries: Z<sub>10</sub>, Z<sub>5</sub>, Z<sub>2</sub>, and Z<sub>1</sub>.

These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges. [[Biography:John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref> Full symmetry of the regular form is '''r20''' and no symmetry is labeled '''a1'''. The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars), and '''i''' when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g10''' subgroup has no degrees of freedom but can be seen as directed edges.

The highest symmetry irregular decagons are '''d10''', an [[Isogonal figure|isogonal]] decagon constructed by five mirrors which can alternate long and short edges, and '''p10''', an [[Isotoxal figure|isotoxal]] decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[Dual polygon|duals]] of each other and have half the symmetry order of the regular decagon.
{{-}}

==Dissection==
{| class=wikitable align=right
![[10-cube|10-cube]] projection
!colspan=4|40 rhomb dissection
|- align=center
|[[File:10-cube t0 A9.svg|100px]]
|[[File:10-gon rhombic dissection-size2.svg|100px]]
|[[File:10-gon rhombic dissection2-size2.svg|100px]]
|[[File:10-gon rhombic dissection3-size2.svg|100px]]
|[[File:10-gon rhombic dissection4-size2.svg|100px]]
|- align=center
|[[File:10-gon rhombic dissection5-size2.svg|100px]]
|[[File:10-gon rhombic dissection6-size2.svg|100px]]
|[[File:10-gon rhombic dissection7-size2.svg|100px]]
|[[File:10-gon rhombic dissection8-size2.svg|100px]]
|[[File:10-gon rhombic dissection9-size2.svg|100px]]
|}
Coxeter states that every [[Zonogon|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular decagon'', ''m''=5, and it can be divided into 10 rhombs, with examples shown below. This decomposition can be seen as 10 of 80 faces in a [[Petrie polygon]] projection plane of the [[5-cube|5-cube]]. A dissection is based on 10 of 30 faces of the [[Rhombic triacontahedron|rhombic triacontahedron]]. The list {{OEIS2C|1=A006245}} defines the number of solutions as 62, with 2 orientations for the first symmetric form, and 10 orientations for the other 6.
{| class=wikitable
|+ Regular decagon dissected into 10 rhombi
|- align=center valign=top
|[[File:5-cube t0.svg|100px]]<BR>[[5-cube|5-cube]]
|[[File:Sun_decagon.svg|100px]]
|[[File:Sun2_decagon.svg|100px]]
|[[File:Dart2 decagon.svg|100px]]
|-
|[[File:Halfsun_decagon.svg|100px]]
|[[File:Dart decagon.svg|100px]]
|[[File:Dart decagon ccw.svg|100px]]
|[[File:Cartwheel_decagon.svg|100px]]
|}
{{-}}

== Skew decagon==
{| class=wikitable align=right width=300
|+ 3 regular skew zig-zag decagons
!{5}#{ }
! {5/2}#{ }
! {5/3}#{ }
|-
|[[File:Regular skew polygon in pentagonal antiprism.png|100px]]
|[[File:Regular skew polygon in pentagrammic antiprism.png|100px]]
|[[File:Regular skew polygon in pentagrammic crossed-antiprism.png|100px]]
|-
|colspan=3|A regular skew decagon is seen as zig-zagging edges of a [[Pentagonal antiprism|pentagonal antiprism]], a [[Pentagrammic antiprism|pentagrammic antiprism]], and a [[Pentagrammic crossed-antiprism|pentagrammic crossed-antiprism]].
|}
A ''skew decagon'' is a [[Skew polygon|skew polygon]] with 10 vertices and edges but not existing on the same plane. The interior of such a decagon is not generally defined. A ''skew zig-zag decagon'' has vertices alternating between two parallel planes.

A ''regular skew decagon'' is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew decagon and can be seen in the vertices and side edges of a [[Pentagonal antiprism|pentagonal antiprism]], [[Pentagrammic antiprism|pentagrammic antiprism]], and [[Pentagrammic crossed-antiprism|pentagrammic crossed-antiprism]] with the same D<sub>5d</sub>, [2<sup>+</sup>,10] symmetry, order 20.

These can also be seen in these four convex polyhedra with [[Icosahedral symmetry|icosahedral symmetry]]. The polygons on the perimeter of these projections are regular skew decagons.
{| class=wikitable width=500
|+ Orthogonal projections of polyhedra on 5-fold axes
|- align=center valign=top
|[[File:Dodecahedron petrie.png|100px]]<br>[[Dodecahedron]]
|[[File:Icosahedron petrie.svg|100px]]<br>[[Icosahedron]]
|[[File:Dodecahedron t1 H3.png|100px]]<br>[[Icosidodecahedron]]
|[[File:Dual dodecahedron t1 H3.png|100px]]<BR>[[Rhombic triacontahedron]]
|}

===Petrie polygons===
The ''regular skew decagon'' is the [[Petrie polygon]] for many higher-dimensional polytopes, shown in these orthogonal projections in various Coxeter planes:<ref>Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223-226.</ref> The number of sides in the Petrie polygon is equal to the [[Coxeter element#Definitions|Coxeter number]], ''h'', for each symmetry family.

{| class=wikitable width=500
!A<sub>9</sub>
!colspan=2|D<sub>6</sub>
!colspan=2|B<sub>5</sub>
|- align=center valign=top
|[[File:9-simplex_t0.svg|100px]]<br>[[9-simplex|9-simplex]]
|[[File:6-cube_t5_B5.svg|100px]]<br>[[5-orthoplex|4<sub>11</sub>]]
|[[File:6-demicube_t0_D6.svg|100px]]<br>[[6-demicube|1<sub>31</sub>]]
|[[File:5-cube_t4.svg|100px]]<br>[[5-orthoplex|5-orthoplex]]
|[[File:5-cube_t0.svg|100px]]<br>[[5-cube|5-cube]]
|}

==See also==
*Decagonal number and centered decagonal number, [[Figurate number|figurate number]]s modeled on the decagon
*[[Decagram (geometry)|Decagram]], a [[Star polygon|star polygon]] with the same vertex positions as the regular decagon

==References==
{{reflist}}

==External links==
*{{MathWorld |urlname=Decagon |title=Decagon}}
*[http://www.mathopenref.com/decagon.html Definition and properties of a decagon] With interactive animation

{{Polygons}}

[[Category:10 (number)]]
[[Category:Constructible polygons]]
[[Category:Elementary shapes]]

{{Sourceattribution|Decagon}}