Regular polygon
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 either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.
Set of convex regular ngons  



Edges and vertices  n  
Schläfli symbol  {n}  
Coxeter–Dynkin diagram  
Symmetry group  D_{n}, order 2n  
Dual polygon  Selfdual  
Area (with side length, s)  
Internal angle  
Internal angle sum  
Inscribed circle diameter  
Circumscribed circle diameter  
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
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}. For n < 3, we have two degenerate cases:
 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.)
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:
 degrees, or equivalently degrees;
 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.
As the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°. As the number of sides increase, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with an infinite number of sides.
Diagonals
For n > 2, the number of diagonals is ; i.e., 0, 2, 5, 9, …, for a triangle, square, pentagon, hexagon, … . The diagonals divide the polygon into 1, 4, 11, 24, … pieces OEIS: A007678.
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.
Points in the plane
For a regular simple ngon with circumradius R and distances d_{i} from an arbitrary point in the plane to the vertices, we have[1]
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[2]^{: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.[3][4]
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
For constructible polygons, algebraic expressions for these relationships exist; see Bicentric polygon#Regular polygons.
The sum of the perpendiculars from a regular ngon's vertices to any line tangent to the circumcircle equals n times the circumradius.[2]^{: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.[2]^{: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} − ns^{2}/4, where s is the side length and R is the circumradius.[2]^{:p. 73}
Dissections
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into or m(m1)/2 parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections mcubes.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. The list OEIS: A006245 gives the number of solutions for smaller polygons.
2m  6  8  10  12  14  16  18  20  24  30  40  50 

Image  
Rhombs  3  6  10  15  21  28  36  45  66  105  190  300 
Area
The area A of a convex regular nsided polygon having side s, circumradius R, apothem a, and perimeter p is given by[6][7]
For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table:[8] (Note that since as [9], the area when is tending to as grows large.)
Number of sides 
Area when side s = 1  Area when circumradius R = 1  Area when apothem a = 1  

Exact  Approximation  Exact  Approximation  As an (approximate) fraction of circumcircle area 
Exact  Approximation  As an (approximate) multiple of incircle area  
n  
3  0.433012702  1.299038105  0.4134966714  5.196152424  1.653986686  
4  1  1.000000000  2  2.000000000  0.6366197722  4  4.000000000  1.273239544 
5  1.720477401  2.377641291  0.7568267288  3.632712640  1.156328347  
6  2.598076211  2.598076211  0.8269933428  3.464101616  1.102657791  
7  3.633912444  2.736410189  0.8710264157  3.371022333  1.073029735  
8  4.828427125  2.828427125  0.9003163160  3.313708500  1.054786175  
9  6.181824194  2.892544244  0.9207254290  3.275732109  1.042697914  
10  7.694208843  2.938926262  0.9354892840  3.249196963  1.034251515  
11  9.365639907  2.973524496  0.9465022440  3.229891423  1.028106371  
12  11.19615242  3  3.000000000  0.9549296586  3.215390309  1.023490523  
13  13.18576833  3.020700617  0.9615188694  3.204212220  1.019932427  
14  15.33450194  3.037186175  0.9667663859  3.195408642  1.017130161  
15  [10]  17.64236291  [11]  3.050524822  0.9710122088  [12]  3.188348426  1.014882824 
16  [13]  20.10935797  3.061467460  0.9744953584  [14]  3.182597878  1.013052368  
17  22.73549190  3.070554163  0.9773877456  3.177850752  1.011541311  
18  25.52076819  3.078181290  0.9798155361  3.173885653  1.010279181  
19  28.46518943  3.084644958  0.9818729854  3.170539238  1.009213984  
20  [15]  31.56875757  [16]  3.090169944  0.9836316430  [17]  3.167688806  1.008306663 
100  795.5128988  3.139525977  0.9993421565  3.142626605  1.000329117  
1000  79577.20975  3.141571983  0.9999934200  3.141602989  1.000003290  
10,000  7957746.893  3.141592448  0.9999999345  3.141592757  1.000000033  
1,000,000  79577471545  3.141592654  1.000000000  3.141592654  1.000000000 
Of all ngons with a given perimeter, the one with the largest area is regular.[18]
Constructible polygon
Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,[19]^{:p. xi} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.[19]^{:pp. 49–50} This led to the question being posed: is it possible to construct all regular ngons with compass and straightedge? If not, which ngons are constructible and which are not?
Carl Friedrich Gauss proved the constructibility of the regular 17gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
 A regular ngon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none).
(A Fermat prime is a prime number of the form ) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.
Equivalently, a regular ngon is constructible if and only if the cosine of its common angle is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.
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
2 < 2q < p, gcd(p, q) = 1
 

Schläfli symbol  {p/q}  
Vertices and Edges  p  
Density  q  
Coxeter diagram  
Symmetry group  Dihedral (D_{p})  
Dual polygon  Selfdual  
Internal angle (degrees) 
[20] 
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:
 Tetragon – {4/2}
 Hexagons – {6/2}, {6/3}
 Octagons – {8/2}, {8/4}
 Enneagon – {9/3}
 Decagons – {10/2}, {10/4}, and {10/5}
 Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6}
Grünbaum {6/2} or 2{3}[21] 
Coxeter 2{3} or {6}[2{3}]{6} 

Doublywound hexagon  Hexagram as a compound of two triangles 
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 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 of two triangles, or hexagram. Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so the hexagram is represented as {6}[2{3}]{6}.[22] More compactly Coxeter also writes 2{n/2}, like 2{3} for a hexagram as compound as alternations of regular evensided polygons, with italics on the leading factor to differentiate it from the coinciding interpretation.[23]
 Many modern geometers, such as Grünbaum (2003),[21] 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
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
 Euclidean tilings by convex regular polygons
 Platonic solid
 Apeirogon – An infinitesided polygon can also be regular, {∞}.
 List of regular polytopes and compounds
 Equilateral polygon
 Carlyle circle
Notes
 Park, PooSung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 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.
 Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 "Math Open Reference". Retrieved 4 Feb 2014.
 "Mathwords".
 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 proc
The expressions for n=16 are obtained by twice applying the tangent halfangle formula to tan(π/4)
 Trigonometric functions
 Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
 Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).
 Kappraff, Jay (2002). Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258. ISBN 9789810247027.
 Are Your Polyhedra the Same as My Polyhedra? Branko Grünbaum (2003), Fig. 3
 Regular polytopes, p.95
 Coxeter, The Densities of the Regular Polytopes II, 1932, p.53
References
 Coxeter, H.S.M. (1948). "Regular Polytopes". Methuen and Co. Cite journal requires
journal=
(help)  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