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| | Im Paige and was born on 25 November 1981. My hobbies are Bus spotting and Sand castle building.<br><br>Here is my homepage: [http://www.marijuanahillbillymagazine.com/mhm/groups/fifa-15-coin-hack/ Fifa 15 Coin Generator] |
| !bgcolor=#e7dcc3 colspan=2|Set of convex regular n-gons
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| |-
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| |align=center colspan=2|
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| [[Image:Regular polygon 3 annotated.svg|60px]][[Image:Regular polygon 4 annotated.svg|60px]][[Image:Regular polygon 5 annotated.svg|60px]][[Image:Regular polygon 6 annotated.svg|60px]]<br>
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| [[Image:Regular polygon 7 annotated.svg|60px]][[Image:Regular polygon 8 annotated.svg|60px]][[Image:Regular polygon 9 annotated.svg|60px]][[Image:Regular polygon 10 annotated.svg|60px]]<br>
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| [[Image:Regular polygon 11 annotated.svg|60px]][[Image:Regular polygon 12 annotated.svg|60px]][[Image:Regular polygon 13 annotated.svg|60px]][[Image:Regular polygon 14 annotated.svg|60px]]<br>
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| [[Image:Regular polygon 15 annotated.svg|60px]][[Image:Regular polygon 16 annotated.svg|60px]][[Image:Regular polygon 17 annotated.svg|60px]][[Image:Regular polygon 18 annotated.svg|60px]]<br>
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| Regular polygons
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| |-
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| |bgcolor=#e7dcc3|[[Edge (geometry)|Edge]]s and [[Vertex (geometry)|vertices]]||''n''
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||{''n''}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]||{{CDD|node_1|n|node}}
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| |-
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| |bgcolor=#e7dcc3|[[Point group|Symmetry group]]||[[Dihedral symmetry|D<sub>n</sub>]], order 2n
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| |-
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| |bgcolor=#e7dcc3|[[Dual polygon]]||Self-dual
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| |-
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| |bgcolor=#e7dcc3|[[Area]]<br /> (with ''s''=side length)||<math>A = \tfrac14ns^2 \cot \frac{\pi}{n}</math>
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| |-
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| |bgcolor=#e7dcc3|[[Internal angle]]||<math>(n-2)\times \frac{180^\circ}{n}</math>
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| |-
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| |bgcolor=#e7dcc3|Internal angle sum||<math>\left(n-2\right)\times 180^\circ</math>
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| |-
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| |bgcolor=#e7dcc3|Properties||[[Convex polygon|convex]], [[Cyclic polygon|cyclic]], [[equilateral]], [[Isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]
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| |}
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| In [[Euclidean geometry]], a '''regular polygon''' is a [[polygon]] that is [[Equiangular polygon|equiangular]] (all angles are equal in measure) and [[equilateral]] (all sides have the same length). Regular polygons may be '''[[Convex and concave polygons|convex]]''' or '''[[Star polygon|star]]'''. In the [[limit (mathematics)|limit]], a sequence of regular polygons with an increasing number of sides becomes a [[circle]], if the [[perimeter]] is fixed, or a regular [[apeirogon]], if the edge length is fixed.
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| ==General properties==
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| ''These properties apply to all regular polygons, whether convex or [[star polygon|star]].''
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| A regular ''n''-sided polygon has [[rotational symmetry]] of order ''n''.
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| All vertices of a regular polygon lie on a common circle (the [[circumscribed circle]]), i.e., they are [[concyclic points]]. That is, a regular polygon is a [[cyclic polygon]].
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| Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or [[incircle]] that is tangent to every side at the midpoint. Thus a regular polygon is a [[tangential polygon]].
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| A regular ''n''-sided polygon can be constructed with [[compass and straightedge]] if and only if the [[odd number|odd]] [[prime number|prime]] factors of ''n'' are distinct [[Fermat prime]]s. See [[constructible polygon]].
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| ===Symmetry===
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| The [[symmetry group]] of an ''n''-sided regular polygon is [[dihedral group]] ''D<sub>n</sub>'' (of order 2''n''): ''D''<sub>2</sub>, [[Dihedral group of order 6|''D''<sub>3</sub>]], [[Examples of groups#A symmetry group|''D''<sub>4</sub>]], ... It consists of the rotations in ''C<sub>n</sub>'', together with [[reflection symmetry]] in ''n'' axes that pass through the center. If ''n'' is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If ''n'' is odd then all axes pass through a vertex and the midpoint of the opposite side.
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| ==Regular convex polygons==
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| All regular [[simple polygon]]s (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also [[Similarity (geometry)|similar]].
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| An ''n''-sided convex regular polygon is denoted by its [[Schläfli symbol]] {''n''}.
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| *[[Henagon]] or monogon {1}: degenerate in [[Euclidean geometry|ordinary space]] (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any [[abstract polytope|abstract polygon]]).
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| *[[Digon]] {2}: a "double line segment": degenerate in [[Euclidean geometry|ordinary space]] (Some authorities do not regard the digon as a true polygon because of this).
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| {|class=wikitable
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| |- align=center
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| |[[File:Regular polygon 3.svg|50px]]<BR>[[Equilateral triangle|Equilateral<BR>triangle]]<BR>{3}
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| |[[File:Regular polygon 4.svg|50px]]<BR>[[square (geometry)|Square]]<BR>{4}
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| |[[File:Regular polygon 5.svg|50px]]<BR>[[Pentagon]]<BR>{5}
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| |[[File:Regular polygon 6.svg|50px]]<BR>[[Hexagon]]<BR>{6}
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| |[[File:Regular polygon 7.svg|50px]]<BR>[[Heptagon]]<BR>or septagon<BR>{7}
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| |[[File:Regular polygon 8.svg|50px]]<BR>[[Octagon]]<BR>{8}
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| |[[File:Regular polygon 9.svg|50px]]<BR>[[Nonagon]] or enneagon<BR>{9}
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| |[[File:Regular polygon 10.svg|50px]]<BR>[[Decagon]]<BR>{10}
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| |- align=center
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| |[[File:Regular polygon 11.svg|50px]]<BR>[[Hendecagon]]<BR>or undecagon<BR>{11}
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| |[[File:Regular polygon 12.svg|50px]]<BR>[[Dodecagon]]<BR>{12}
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| |[[File:Regular polygon 13.svg|50px]]<BR>[[Tridecagon]]<BR>{13}
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| |[[File:Regular polygon 14.svg|50px]]<BR>[[Tetradecagon]]<BR>{14}
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| |[[File:Regular polygon 15.svg|50px]]<BR>[[Pentadecagon]]<BR>{15}
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| |[[File:Regular polygon 16.svg|50px]]<BR>[[Hexadecagon]]<BR>{16}
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| |[[File:Regular polygon 17.svg|50px]]<BR>[[Heptadecagon]]<BR>{17}
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| |[[File:Regular polygon 18.svg|50px]]<BR>[[Octadecagon]]<BR>{18}
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| |[[File:Regular polygon 19.svg|50px]]<BR>[[Enneadecagon]]<BR>{19}
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| |- align=center
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| |[[File:Regular polygon 20.svg|50px]]<BR>[[Icosagon]]<BR>{20}
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| |[[File:Regular polygon 30.svg|50px]]<BR>[[Triacontagon]]<BR>{30}
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| |[[File:Regular polygon 40.svg|50px]]<BR>[[Tetracontagon]]<BR>{40}
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| |[[File:Regular polygon 50.svg|50px]]<BR>[[Pentacontagon]]<BR>{50}
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| |[[File:Regular polygon 60.svg|50px]]<BR>[[Hexacontagon]]<BR>{60}
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| |[[File:Regular polygon 70.svg|50px]]<BR>[[Heptacontagon]]<BR>{70}
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| |[[File:Regular polygon 80.svg|50px]]<BR>[[Octacontagon]]<BR>{80}
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| |[[File:Regular polygon 90.svg|50px]]<BR>[[Enneacontagon]]<BR>{90}
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| |[[File:Regular polygon 100.svg|50px]]<BR>[[Hectogon]]<BR>{100}
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| |}
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| In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of [[uniform polyhedra]] must be regular and the faces will be described simply as triangle, square, pentagon, etc.
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| | |
| ===Angles===
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| For a regular convex ''n''-gon, each interior angle has a measure of:
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| :<math>\left(1-\frac{2}{n}\right)\times 180</math> (or equally of <math>(n-2)\times \frac{180}{n}</math> ) degrees,
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| :or <math>\frac{(n-2)\pi}{n}</math> radians,
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| :or <math>\frac{(n-2)}{2n}</math> full [[Turn (geometry)|turns]],
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| and each [[Internal and external angle|exterior angle]] (i.e. [[supplementary angle|supplementary]] to the interior angle) has a measure of <math>\tfrac{360}{n}</math> degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.
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| ===Diagonals===
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| For ''n'' > 2 the number of [[diagonal]]s is <math>\tfrac{n (n-3)}{2}</math>, i.e., 0, 2, 5, 9, ... for a triangle, quadrilateral, pentagon, hexagon, .... The diagonals divide the polygon into 1, 4, 11, 24, ... pieces.
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| For a regular ''n''-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals ''n''.
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| <!--===Height===
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| ISTR that a formula exists for the height of a regular polygon. If you know it, please add it here. Well it's just 2 times apothem for even n, or apothem + radius for odd n. Is it worth recording?-->
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| ===Interior points===
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| For a regular ''n''-gon, the sum of the perpendicular distances from any interior point to the ''n'' sides is ''n'' times the [[apothem]] (the apothem being the distance from the center to any side). This is a generalization of [[Viviani's theorem]] for the ''n''=3 case.<ref>Pickover, Clifford A, ''The Math Book'', Sterling, 2009: p. 150</ref><ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>
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| ===Circumradius===
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| The [[circumradius]] from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the [[apothem]] ''a'' by
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| :<math>r=\frac{s}{2 \sin{ \frac{\pi}{n} }} = \frac{a}{\cos{ \frac{\pi}{n} }}</math>
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| ===Area===<!--This section is linked from [[Truncated icosahedron]]-->
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| [[Image:PolygonParameters.png|thumb|right|Regular polygon with ''n'' = 5: [[pentagon]] with [[side (geometry)|side]] ''s'', [[circumradius]] ''r'' and [[apothem]] ''a'']]
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| The area ''A'' of a convex regular ''n''-sided polygon having [[side (geometry)|side]] ''s'', [[circumradius]] ''r'', [[apothem]] ''a'', and [[perimeter]] ''p'' is given by<ref>{{cite web |url=http://www.mathwords.com/a/area_regular_polygon.htm |title=Mathworlds}}</ref>
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| :<math>A= \tfrac{1}{2}nsa = \tfrac{1}{2}pa = \tfrac{1}{4}ns^2\cot{\tfrac{\pi}{n}} = na^2\tan{\tfrac{\pi}{n}} = \tfrac{1}{2}nr^2\sin{\tfrac{2\pi}{n}}</math>
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| For regular polygons with side ''s''=1, circumradius ''r'' =1, or apothem ''a''=1, this produces the following table:<ref>Results for ''r''=1 and ''a''=1 obtained with [[Maple (software)|Maple]], using function definition:<p><code>f := proc (n)<br /> options operator, arrow;<br />[<br />[convert((1/4)*n*cot(Pi/n), radical), convert((1/4)*n*cot(Pi/n), float)],<br /> [convert((1/2)*n*sin(2*Pi/n), radical), convert((1/2)*n*sin(2*Pi/n), float), convert((1/2)*n*sin(2*Pi/n)/Pi, float)],<br /> [convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]<br />]<br /> end proc</code><p>The expressions for ''n''=16 are obtained by twice applying the [[tangent half-angle formula]] to tan(π/4)</ref>
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| {|class=wikitable style="text-align:center"
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| |-
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| !rowspan="2"| Number of sides
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| !rowspan="2"| Name of polygon
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| !style="background:#FF8080" colspan="2"| Area when side ''s''=1
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| !style="background:#80FF80" colspan="3"| Area when circumradius ''r''=1
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| !style="background:#8080FF" colspan="3"| Area when apothem ''a''=1
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| |-
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| !Exact
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| !Approximate
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| !Exact
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| !Approximate
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| !Approximate as fraction of circle
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| !Exact
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| !Approximate
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| !Approximate as fraction of circle
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| |-
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| |''n''
| |
| |regular ''n''-gon || <math>\tfrac{n}{4}\cot{\tfrac{\pi}{n}}</math> || || <math>\tfrac{n}{2}\sin{\tfrac{2\pi}{n}}</math> || || <math>\tfrac{n}{2\pi}\sin{\tfrac{2\pi}{n}}</math> || <math>n \tan{\tfrac{\pi}{n}}</math> || || <math>\tfrac{n}{\pi}\tan{\tfrac{\pi}{n}}</math>
| |
| |-
| |
| |3
| |
| |[[equilateral triangle]] || {{math|{{radical|3}}/4}} || 0.433012702 || {{math|3{{radical|3}}/4}} || 1.299038105 || 0.4134966714 || {{math|3{{radical|3}}}} || 5.196152424 || 1.653986686
| |
| |-
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| |4
| |
| |[[square (geometry)|square]] || 1 || 1.000000000 || 2 || 2.000000000 || 0.6366197722 || 4 || 4.000000000 || 1.273239544
| |
| |-
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| |5
| |
| |regular [[pentagon]] || {{math|1/4{{radical|25+10{{radical|5}}}}}} || 1.720477401 || {{math|5/4{{radical|(5+{{radical|5}})/2}}}} || 2.377641291 || 0.7568267288 || {{math|5{{radical|5-2{{radical|5}}}}}} || 3.632712640 || 1.156328347
| |
| |-
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| |6
| |
| |regular [[hexagon]] || {{math|3{{radical|3}}/2}} || 2.598076211 || {{math|3{{radical|3}}/2}} || 2.598076211 || 0.8269933428 || {{math|2{{radical|3}}}} || 3.464101616 || 1.102657791
| |
| |-
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| |7
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| |regular [[heptagon]] || || 3.633912444 || || 2.736410189 || 0.8710264157 || || 3.371022333 || 1.073029735
| |
| |-
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| |8
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| |regular [[octagon]] || {{math|2+2{{radical|2}}}} || 4.828427125 || {{math|2{{radical|2}}}} || 2.828427125 || 0.9003163160 || {{math|8({{radical|2}}-1)}} || 3.313708500 || 1.054786175
| |
| |-
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| |9
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| |regular [[nonagon]] || || 6.181824194 || || 2.892544244 || 0.9207254290 || || 3.275732109 || 1.042697914
| |
| |-
| |
| |10
| |
| |regular [[decagon]] || {{math|5/2{{radical|5+2{{radical|5}}}}}} || 7.694208843 || {{math|5/2{{radical|(5-{{radical|5}})/2}}}} || 2.938926262 || 0.9354892840 || {{math|2{{radical|25-10{{radical|5}}}}}} || 3.249196963 || 1.034251515
| |
| |-
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| |11
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| |regular [[hendecagon]] || || 9.365639907 || || 2.973524496 || 0.9465022440 || || 3.229891423 || 1.028106371
| |
| |-
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| |12
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| |regular [[dodecagon]] || {{math|6+3{{radical|3}}}} || 11.19615242 || 3 || 3.000000000 || 0.9549296586 || {{math|12(2-{{radical|3}})}} || 3.215390309 || 1.023490523
| |
| |-
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| |13
| |
| |regular [[triskaidecagon]] || || 13.18576833 || || 3.020700617 || 0.9615188694 || || 3.204212220 || 1.019932427
| |
| |-
| |
| |14
| |
| |regular [[tetradecagon]] || || 15.33450194 || || 3.037186175 || 0.9667663859 || || 3.195408642 || 1.017130161
| |
| |-
| |
| |15
| |
| |regular [[pentadecagon]] || || 17.64236291 || || 3.050524822 || 0.9710122088 || || 3.188348426 || 1.014882824
| |
| |-
| |
| |16
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| |regular [[hexadecagon]] || {{math| 4 (1+{{radical|2}}+{{radical|2 (2+{{radical|2}})}})}} || 20.10935797 || {{math|4{{radical|2-{{radical|2}}}}}} || 3.061467460 || 0.9744953584 || {{math| 16 (1+{{radical|2}})({{radical|2 (2-{{radical|2}})}}-1)}} || 3.182597878 || 1.013052368
| |
| |-
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| |17
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| |regular [[heptadecagon]] || || 22.73549190 || || 3.070554163 || 0.9773877456 || || 3.177850752 || 1.011541311
| |
| |-
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| |18
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| |regular [[octadecagon]] || || 25.52076819 || || 3.078181290 || 0.9798155361 || || 3.173885653 || 1.010279181
| |
| |-
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| |19
| |
| |regular [[enneadecagon]] || || 28.46518943 || || 3.084644958 || 0.9818729854 || || 3.170539238 || 1.009213984
| |
| |-
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| |20
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| |regular [[icosagon]] || {{math| 5 (1+{{radical|5}}+{{radical|5+2{{radical|5}}}}) }} || 31.56875757 || {{math|5/2 ({{radical|5}}-1)}} || 3.090169944 || 0.9836316430 || {{math| 20 (1+{{radical|5}}-{{radical|5+2{{radical|5}}}}) }} || 3.167688806 || 1.008306663
| |
| |-
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| |100
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| |regular hectagon || || 795.5128988 || || 3.139525977 || 0.9993421565 || || 3.142626605 || 1.000329117
| |
| |-
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| |1000
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| |regular [[chiliagon]] || || 79577.20975 || || 3.141571983 || 0.9999934200 || || 3.141602989 || 1.000003290
| |
| |-
| |
| |10000
| |
| |regular myriagon || || 7957746.893 || || 3.141592448 || 0.9999999345 || || 3.141592757 || 1.000000033
| |
| |-
| |
| |1,000,000
| |
| |regular [[megagon]] || || {{formatnum:79577471545.685}} || || 3.141592654 || 1.000000000 || || 3.141592654 || 1.000000000
| |
| |}
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| | |
| Of all ''n''-gons with a given perimeter, the one with the largest area is regular.<ref>Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>
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| =={{anchor|Skew regular polygons}}Regular skew polygons==
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| {|class=wikitable align=right width=400
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| |- valign=top
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| |[[File:Cube petrie polygon sideview.png|160px]]<BR>The [[cube]] contains a skew regular [[hexagon]], seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis.
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| |[[File:Antiprism17.jpg|240px]]<BR>The zig-zagging side edges of a ''n''-[[antiprism]] represent a regular skew 2''n''-gon, as shown in this 17-gonal antiprism.
| |
| |}
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| A ''regular [[skew polygon]]'' in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform [[antiprism]]. All edges and internal angles are equal.
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| {|class=wikitable width=480
| |
| |[[File:Petrie polygons.png|480px]]<BR>The [[Platonic solid]]s (the [[tetrahedron]], [[cube]], [[octahedron]], [[dodecahedron]], and icosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively.
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| |}
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| More generally ''regular skew polygons'' can be defined in ''n''-space. Examples include the [[Petrie polygon]]s, polygonal paths of edges that divide a [[regular polytope]] into two halves, and seen as a regular polygon in orthogonal projection.
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| In the infinite limit ''regular skew polygons'' become skew [[apeirogon]]s.
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| {{-}}
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| ==Regular star polygons==
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| [[Image:Pentagram green.svg|thumb|150px|right|A pentagram {5/2}]]
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| A non-convex regular polygon is a regular [[star polygon]]. The most common example is the [[pentagram]], which has the same vertices as a [[pentagon]], but connects alternating vertices.
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| | |
| For an ''n''-sided star polygon, the [[Schläfli symbol]] is modified to indicate the ''density'' or "starriness" ''m'' of the polygon, as {''n''/''m''}. If ''m'' is 2, for example, then every second point is joined. If ''m'' is 3, then every third point is joined. The boundary of the polygon winds around the center ''m'' times.
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| | |
| The (non-degenerate) regular stars of up to 12 sides are:
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| *[[Pentagram]] – {5/2}
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| *[[Heptagram]] – {7/2} and {7/3}
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| *[[Octagram]] – {8/3}
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| *[[Enneagram (geometry)|Enneagram]] – {9/2} and {9/4}
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| *[[Decagram (geometry)|Decagram]] – {10/3}
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| *[[Hendecagram]] – {11/2}, {11/3}, {11/4} and {11/5}
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| *[[Dodecagram]] – {12/5}
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| ''m'' and ''n'' must be [[co-prime]], or the figure will degenerate.
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| The degenerate regular stars of up to 12 sides are:
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| *[[Hexagram]] – {6/2}
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| *[[Octagram]] – {8/2}
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| *[[Enneagram (geometry)|Enneagram]] – {9/3}
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| *[[Decagram (geometry)|Decagram]] – {10/2} and {10/4}
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| *[[Dodecagram]] – {12/2}, {12/3} and {12/4}
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| Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example {6/2} may be treated in either of two ways:
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| *For much of the 20th century (see for example {{harvtxt|Coxeter|1948}}), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbors two steps away, to obtain the regular [[compound polygon|compound]] of two triangles, or [[hexagram]].
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| *Many modern geometers, such as Grünbaum (2003), regard this as incorrect. They take the /2 to indicate moving two places around the {6} at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories of [[abstract polytope]]s, but it also more closely copies the way in which Poinsot (1809) created his star polygons – by taking a single length of wire and bending it at successive points through the same angle until the figure closed.
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| | |
| ==Duality of regular polygons ==
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| {{see also|Self-dual polyhedra}}
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| All regular polygons are self-dual to congruency, and for odd ''n'' they are self-dual to identity.
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| In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.
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| ==Regular polygons as faces of polyhedra==
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| A [[uniform polyhedron]] has regular polygons as faces, such that for every two vertices there is an [[isometry]] mapping one into the other (just as there is for a regular polygon).
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| | |
| A [[quasiregular polyhedron]] is a uniform polyhedron which has just two kinds of face alternating around each vertex.
| |
| | |
| A [[regular polyhedron]] is a uniform polyhedron which has just one kind of face.
| |
| | |
| The remaining (non-uniform) [[convex polyhedra]] with regular faces are known as the [[Johnson solids]].
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| | |
| A polyhedron having regular triangles as faces is called a [[deltahedron]].
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| ==See also==
| |
| *[[Tiling by regular polygons]]
| |
| *[[Platonic solids]]
| |
| *[[Apeirogon]] – An infinite-sided polygon can also be regular, {∞}.
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| *[[List of regular polytopes]]
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| *[[Equilateral polygon]]
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| *[[Carlyle circle]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Cite journal |authorlink=Coxeter |first=H.S.M. |last=Coxeter |title=Regular Polytopes |publisher=Methuen and Co. |year=1948 |ref=harv |postscript=<!--None-->}}
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| *Grünbaum, B.; Are your polyhedra the same as my polyhedra?, ''Discrete and comput. geom: the Goodman-Pollack festschrift'', Ed. Aronov et al., Springer (2003), pp. 461–488.
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| *[[Louis Poinsot|Poinsot, L.]]; Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' '''9''' (1810), pp. 16–48.
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| ==External links==
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| *{{mathworld |urlname=RegularPolygon |title=Regular polygon}}
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| *[http://www.mathopenref.com/polygonregular.html Regular Polygon description] With interactive animation
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| *[http://www.mathopenref.com/polygonincircle.html Incircle of a Regular Polygon] With interactive animation
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| *[http://www.mathopenref.com/polygonregulararea.html Area of a Regular Polygon] Three different formulae, with interactive animation
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| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1056&bodyId=1245 Renaissance artists' constructions of regular polygons] at [http://mathdl.maa.org/convergence/1/ Convergence]
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| {{Polygons}}
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| {{DEFAULTSORT:Regular Polygon}}
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| [[Category:Polygons]]
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| [[de:Polygon#Regelmäßige Polygone]]
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