- Regular polygon
A

**regular polygon**is apolygon which is equiangular (all angles are congruent) andequilateral (all sides have the same length). Regular polygons may be**convex**or**star**.**General properties**These properties apply to both convex and star regular polygons.

All vertices of a regular polygon lie on a common circle, i.e., they are

concyclic points , i.e., every regular polygon has acircumscribed circle .Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or

incircle .A regular "n"-sided polygon can be constructed with

compass and straightedge if and only if the odd prime factors of "n" are distinctFermat prime s. Seeconstructible polygon .**ymmetry**The

symmetry group of an "n"-sided regular polygon isdihedral group "D_{n}" (of order 2"n"): "D"_{2}, "D"_{3}, "D"_{4},... It consists of the rotations in "C_{n}" (there isrotational symmetry of order "n"), together withreflection 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.**Regular convex polygons**All regular

simple polygon s (a simple polygon is one which does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.An "n"-sided convex regular polygon is denoted by its

Schläfli symbol {"n"}.*

Henagon or monogon: degenerate in ordinary space {1}

*Digon : a "double line segment" - degenerate in ordinary space {2}

*Equilateral triangle {3}

* Square {4}

* Regularpentagon {5}

* Regularhexagon {6}

* Regularheptagon {7}

* Regularoctagon {8}

* Regularenneagon or nonagon {9}

* Regulardecagon {10}

* Regularhendecagon {11}

* Regulardodecagon {12}

* Regulartriskaidecagon {13}

* Regulartetrakaidecagon {14}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.**Angles**For a regular convex "n"-gon, each interior angle has a measure of:

:$(1-frac\{2\}\{n\})\; imes\; 180$ (or equally of $(n-2)\; imes\; frac\{180\}\{n\}$ ) degrees,

:or $frac\{(n-2)pi\}\{n\}$ radians,

:or $frac\{(n-2)\}\{2n\}$ full turns,

and each exterior angle (supplementary to the interior angle) has a measure of $frac\{360\}\{n\}$ degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

**Diagonals**For $n\; >\; 2$ the number of

diagonal s is $frac\{n\; (n-3)\}\{2\}$, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.**Area**The area A of a convex regular "n"-sided polygon is:

in degrees:$A=frac\{nt^2\}\{4\; an(frac\{180\}\{n\})\}$,

or in radians:$A=frac\{nt^2\}\{4\; an(frac\{pi\}\{n\})\}$,

where "t" is the length of a side.

If the radius, or length of the segment joining the center to the vertex is known, the area is:

in degrees:$A=frac\{nr^2sin(frac\{360\}\{n\})\}\{2\}$

or in radians:$A=frac\{nr^2sin(frac\{2\; pi\}\{n\})\}\{2\}$,

where "r" is the radius

Also, the area is half the perimeter multiplied by the length of the

apothem , "a", (the line drawn from the centre of the polygon perpendicular to a side). That is "A" = "a.n.t"/2, as the length of the perimeter is "n.t", or more simply 1/2 "p.a".For sides "t"=1 this gives:

in degrees:$frac\{n\}\{4\; an(frac\{180\}\{n\})\}$

or in radians ("n" not equal to 2):$\{frac\{n\}\{4\; cot(pi/n)$

with the following values:

The amounts that the areas are less than those of circles with the same

perimeter , are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing "n" to the limit π/12).**Regular star polygons**A non-convex regular polygon is a regular

star polygon . The most common example is thepentagram , which has the same vertices as apentagon , but connects alternating vertices.For an "n"-sided star polygon, the

Schläfli symbol is modified to indicate the '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 centre "m" times, and "m" is sometimes called the**density**of the polygon.Examples:

*Pentagram - {5/2}

*Heptagram - {7/2} and {7/3}

*Octagram - {8/3}

*Enneagram - {9/2} and {9/4}

***Decagram**- {10/3}

*Hendecagram - {11/2}, {11/3}, {11/4}, {11/5}"m" and "n" must be

co-prime , or the figure will degenerate. 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:

* 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 neighbours two steps away, to obtain the regular compound of two triangles, orhexagram .

* 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 ofabstract 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.**elf-dual polygons**All

regular polygon s are self-dual, with equal number of vertices and edges.In addition the regular

star polygon s and regular star figures (compounds), being composed of regular polygons, are also self-dual.**Regular polygons as faces of polyhedra**A

uniform polyhedron is apolyhedron with regular polygons as faces such that for every two vertices there is anisometry mapping one into the other (just as there is for a regular polygon).The remaining

convex polyhedra with regular faces are known as theJohnson solids .**References***citation|authorlink=Coxeter|first=H. S. M.|last= Coxeter | title=Regular Polytopes | publisher=Methuen and Co.|year=1948

*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.

*Poinsot, L.; Memoire sur les polygones et polyèdres. "J. de l'École Polytechnique"**9**(1810), pp. 16-48.**ee also***

Tiling by regular polygons

*Platonic solids

*Apeirogon - An infinite-sided polygon can also be regular, {∞}.

*List of regular polytopes

*Equilateral polygon **External links***mathworld | urlname = RegularPolygon | title = Regular polygon

* [*http://www.mathopenref.com/polygonregular.html Regular Polygon description*] With interactive animation

* [*http://www.mathopenref.com/polygonincircle.html Incircle of a Regular Polygon*] With interactive animation

* [*http://www.mathopenref.com/polygonregulararea.html Area of a Regular Polygon*] Three different formulae, with interactive animation

* [*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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