REGULAR POLYGON

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{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Set of regular p-gons
|-
|align=center colspan=2|
Example: A regular pentagon, {5}
|-
|bgcolor=#e7dcc3|Edges and vertices||p
|-
|bgcolor=#e7dcc3|Schläfli symbol||{p}
|-
|bgcolor=#e7dcc3|Coxeter–Dynkin diagram||
|-
|bgcolor=#e7dcc3|Symmetry group||Dihedral (D_p)
|-
|bgcolor=#e7dcc3|Dual polyhedron||Self-dual
|-
|bgcolor=#e7dcc3|Area
(with ''t''=edge length)||
|-
|bgcolor=#e7dcc3|Internal angle
(degrees)||
ight) imes 180
|}
A 'regular polygon' is a polygon which is equiangular (all angles are congruent) and equilateral (all sides have the same length).

Contents

Regular convex polygons

Properties

Area

Regular star polygons

Symmetry

Regular polygons as faces of polyhedra

See also

External links

Regular convex polygons

All regular simple polygons (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''}.

★ Regular digon: degenerate, a "double line segment" {2}

★ Equilateral triangle {3}

★ Square {4}

★ Regular pentagon {5}

★ Regular hexagon {6}

★ Regular heptagon {7}

★ Regular octagon {8}

★ Regular decagon {10}

★ Regular dodecagon {12}
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.

Properties

Each angle of a regular ''n''-gon has a measure of (or equally of ) degrees.
Alternately, the internal angle(s) of a regular ''n''-gon is radians ( or turns).
All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points, i.e., every regular polygon has a circumscribed circle.
A regular ''n''-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of ''n'' are distinct Fermat primes. See constructible polygon.
For $n\; >\; 2$ the number of diagonals is , i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.

Area

The area of a regular ''n''-sided polygon is
:
where
:''t'' is the length of a side.
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 easier 1/2 p a.
For ''t''=1 this gives
:
with the following values:
{| class=wikitable
|-
!Sides!!Name!!Exact area!!Approximate area
|- align="right"
|3
|align="left"|equilateral triangle||||0.433
|- align="right"
|4
|align="left"|square||1||1.000
|- align="right"
|5
|align="left"|regular-pentagon||||1.720
|- align="right"
|6
|align="left"|regular-hexagon||||2.598
|- align="right"
|7
|align="left"|regular-heptagon|| ||3.634
|- align="right"
|8
|align="left"|regular-octagon||$2\; +\; 2\; sqrt\{2\}$||4.828
|- align="right"
|9
|align="left"|regular-enneagon|| ||6.182
|- align="right"
|10
|align="left"|regular-decagon||||7.694
|- align="right"
|11
|align="left"|regular-hendecagon|| ||9.366
|- align="right"
|12
|align="left"|regular-dodecagon||$6+3sqrt\{3\}$||11.196
|- align="right"
|13
|align="left"|regular-triskaidecagon|| ||13.186
|- align="right"
|14
|align="left"|regular-tetradecagon|| ||15.335
|- align="right"
|15
|align="left"|regular-pentadecagon|| ||17.642
|- align="right"
|16
|align="left"|regular-hexadecagon|| ||20.109
|- align="right"
|17
|align="left"|regular-heptadecagon|| ||22.735
|- align="right"
|18
|align="left"|regular-octadecagon|| ||25.521
|- align="right"
|19
|align="left"|regular-enneadecagon|| ||28.465
|- align="right"
|20
|align="left"|regular-icosagon|| ||31.569
|- align="right"
|100
|align="left"|regular-hectagon|| ||795.513
|- align="right"
|1000
|align="left"|regular-chiliagon|| ||79577.210
|- align="right"
|10000
|align="left"|regular-myriagon|| ||7957746.893
|}
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 called a star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, 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.
Examples:

★ Pentagram - {5/2}

★ Heptagram - {7/2} and {7/3}

★ 'Octagram' - {8/3}

★ Enneagram - {9/2} and {9/4}

★ 'Decagram' - {10/3}

Symmetry

The symmetry group of an ''n''-sided regular polygon is dihedral group ''D_n'' (of order 2''n''): ''D''₂, ''D''₃, ''D''₄,... It consists of the rotations in ''C_n'' (there is rotational symmetry of order ''n''), 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.

Regular polygons as faces of polyhedra

A uniform polyhedron is a polyhedron with 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).
The remaining convex polyhedra with regular faces are known as the Johnson solids.

See also

★ tiling by regular polygons

★ Platonic solids

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

★ List of regular polytopes

External links

★

★ Regular Polygon description With interactive animation

★ Incircle of a Regular Polygon With interactive animation

★ Area of a Regular Polygon Three different formulae, with interactive animation

★ Renaissance artists' constructions of regular polygons at Convergence