Regular polygon
Set of convex regular ngons  

Edges and vertices  n 
Schläfli symbol  {n} 
Coxeter–Dynkin diagram  Template:CDD 
Symmetry group  D_{n}, order 2n 
Dual polygon  Selfdual 
Area (with s=side length) 

Internal angle  
Internal angle sum  
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star. In the 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.
General properties
These properties apply to all regular polygons, whether convex or star.
A regular nsided polygon has rotational symmetry of order n.
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.
Together with the property of equallength 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.
A regular nsided 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.
Symmetry
The symmetry group of an nsided regular polygon is dihedral group D_{n} (of order 2n): D_{2}, D_{3}, D_{4}, ... It consists of the rotations in C_{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 convex polygons
All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.
An nsided convex regular polygon is denoted by its Schläfli symbol {n}.
 Monogon {1}: degenerate in 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 polygon).
 Digon {2}: a "double line segment": degenerate in ordinary space (Some authorities do not regard the digon as a true polygon because of this).
Equilateral triangle {3} 
Square {4} 
Pentagon {5} 
Hexagon {6} 
Heptagon or septagon {7} 
Octagon {8} 
Nonagon or enneagon {9} 
Decagon {10}  
Hendecagon or undecagon {11} 
Dodecagon {12} 
Tridecagon {13} 
Tetradecagon {14} 
Pentadecagon {15} 
Hexadecagon {16} 
Heptadecagon {17} 
Octadecagon {18} 
Enneadecagon {19} 
Icosagon {20} 
Triacontagon {30} 
Tetracontagon {40} 
Pentacontagon {50} 
Hexacontagon {60} 
Heptacontagon {70} 
Octacontagon {80} 
Enneacontagon {90} 
Hectogon {100} 
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 ngon, each interior angle has a measure of:
 (or equally of ) degrees,
 or radians,
 or full turns,
and each exterior angle (i.e. supplementary to the interior angle) has a measure of 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 diagonals is , i.e., 0, 2, 5, 9, ... for a triangle, quadrilateral, pentagon, hexagon, .... The diagonals divide the polygon into 1, 4, 11, 24, ... pieces.
For a regular ngon inscribed in a unitradius 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.
Interior points
For a regular ngon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem^{[1]}^{:p. 72} (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n=3 case.^{[2]}^{[3]}
Circumradius
The circumradius R 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
The sum of the perpendiculars from a regular ngon's vertices to any line tangent to the circumcircle equals n times the circumradius.^{[1]}^{:p. 73}
The sum of the squared distances from the vertices of a regular ngon to any point on its circumcircle equals 2nR^{2} where R is the circumradius.^{[1]}^{:p.73}
The sum of the squared distances from the midpoints of the sides of a regular ngon to any point on the circumcircle is 2nR^{2} — (na^{2})/4, where a is the side length and R is the circumradius.^{[1]}^{:p. 73}
Area
The area A of a convex regular nsided polygon having side s, circumradius R, apothem a, and perimeter p is given by^{[4]}^{[5]}
For regular polygons with side s=1, circumradius R =1, or apothem a=1, this produces the following table:^{[6]}
Number of sides  Name of polygon  Area when side s=1  Area when circumradius R=1  Area when apothem a=1  

Exact  Approximate  Exact  Approximate  Approximate as fraction of circle  Exact  Approximate  Approximate as fraction of circle  
n  regular ngon  
3  equilateral triangle  Template:Radical/4  0.433012702  3Template:Radical/4  1.299038105  0.4134966714  3Template:Radical  5.196152424  1.653986686 
4  square  1  1.000000000  2  2.000000000  0.6366197722  4  4.000000000  1.273239544 
5  regular pentagon  1/4Template:Radical  1.720477401  5/4Template:Radical  2.377641291  0.7568267288  5Template:Radical  3.632712640  1.156328347 
6  regular hexagon  3Template:Radical/2  2.598076211  3Template:Radical/2  2.598076211  0.8269933428  2Template:Radical  3.464101616  1.102657791 
7  regular heptagon  3.633912444  2.736410189  0.8710264157  3.371022333  1.073029735  
8  regular octagon  2+2Template:Radical  4.828427125  2Template:Radical  2.828427125  0.9003163160  8(Template:Radical1)  3.313708500  1.054786175 
9  regular nonagon  6.181824194  2.892544244  0.9207254290  3.275732109  1.042697914  
10  regular decagon  5/2Template:Radical  7.694208843  5/2Template:Radical  2.938926262  0.9354892840  2Template:Radical  3.249196963  1.034251515 
11  regular hendecagon  9.365639907  2.973524496  0.9465022440  3.229891423  1.028106371  
12  regular dodecagon  6+3Template:Radical  11.19615242  3  3.000000000  0.9549296586  12(2Template:Radical)  3.215390309  1.023490523 
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  regular hexadecagon  4 (1+Template:Radical+Template:Radical)  20.10935797  4Template:Radical  3.061467460  0.9744953584  16 (1+Template:Radical)(Template:Radical1)  3.182597878  1.013052368 
17  regular heptadecagon  22.73549190  3.070554163  0.9773877456  3.177850752  1.011541311  
18  regular octadecagon  25.52076819  3.078181290  0.9798155361  3.173885653  1.010279181  
19  regular enneadecagon  28.46518943  3.084644958  0.9818729854  3.170539238  1.009213984  
20  regular icosagon  5 (1+Template:Radical+Template:Radical)  31.56875757  5/2 (Template:Radical1)  3.090169944  0.9836316430  20 (1+Template:RadicalTemplate:Radical)  3.167688806  1.008306663 
100  regular hectagon  795.5128988  3.139525977  0.9993421565  3.142626605  1.000329117  
1000  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  79,577,471,545.685  3.141592654  1.000000000  3.141592654  1.000000000 
Of all ngons with a given perimeter, the one with the largest area is regular.^{[7]}
{{safesubst:#invoke:anchormain}}Regular skew polygons
The cube contains a skew regular hexagon, seen as 6 red edges zigzagging between two planes perpendicular to the cube's diagonal axis. 
The zigzagging side edges of a nantiprism represent a regular skew 2ngon, as shown in this 17gonal antiprism. 
A regular skew polygon in 3space can be seen as nonplanar paths zigzagging between two parallel planes, defined as the sideedges of a uniform antiprism. All edges and internal angles are equal.
The Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively. 
More generally regular skew polygons can be defined in nspace. Examples include the Petrie polygons, polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection.
In the infinite limit regular skew polygons become skew apeirogons.
Regular star polygons
A nonconvex 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.
For an nsided 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.
The (nondegenerate) regular stars of up to 12 sides are:
 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} and {11/5}
 Dodecagram – {12/5}
m and n must be coprime, or the figure will degenerate.
The degenerate regular stars of up to 12 sides are:
 Hexagram – {6/2}
 Octagram – {8/2}
 Enneagram – {9/3}
 Decagram – {10/2} and {10/4}
 Dodecagram – {12/2}, {12/3} and {12/4}
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 Template:Harvtxt), 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 of two triangles, or hexagram.
 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 "doublewound" 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 polytopes, 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.
Duality of regular polygons
{{#invoke:see alsoseealso}} All regular polygons are selfdual to congruency, and for odd n they are selfdual to identity.
In addition, the regular star figures (compounds), being composed of regular polygons, are also selfdual.
Regular polygons as faces of polyhedra
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).
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 (nonuniform) convex polyhedra with regular faces are known as the Johnson solids.
A polyhedron having regular triangles as faces is called a deltahedron.
See also
 Tiling by regular polygons
 Platonic solids
 Apeirogon – An infinitesided polygon can also be regular, {∞}.
 List of regular polytopes
 Equilateral polygon
 Carlyle circle
Notes
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
 ↑ Pickover, Clifford A, The Math Book, Sterling, 2009: p. 150
 ↑ Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", The College Mathematics Journal 37(5), 2006, pp. 390–391.
 ↑ Template:Cite web
 ↑ Template:Cite web
 ↑ Results for R=1 and a=1 obtained with Maple, using function definition:
f := proc (n)
options operator, arrow;
[
[convert((1/4)*n*cot(Pi/n), radical), convert((1/4)*n*cot(Pi/n), float)],
[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)],
[convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]
]
end procThe expressions for n=16 are obtained by twice applying the tangent halfangle formula to tan(π/4)
 ↑ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
References
 {{#invoke:Citation/CS1citation
CitationClass=journal }}
 Grünbaum, B.; Are your polyhedra the same as my polyhedra?, Discrete and comput. geom: the GoodmanPollack 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.
External links
 Weisstein, Eric W., "Regular polygon", MathWorld.
 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
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