Set of convex regular ngons  

Edges and vertices  n 
Schläfli symbol  {n} 
CoxeterDynkin 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 
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.
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.
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.
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:
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.
For a regular convex ngon, each interior angle has a measure of:
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 2? 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.
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 .
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.
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]}
For higher powers of distances from an arbitrary point in the plane to the vertices of a regular gon, if
then^{[2]}
and
where is a positive integer less than .
If is the distance from an arbitrary point in the plane to the centroid of a regular gon with circumradius , then ^{[2]}
where = 1,2,..., 1.
For a regular ngon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem^{[3]}^{: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.^{[4]}^{[5]}
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.^{[3]}^{: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.^{[3]}^{: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}  , where s is the side length and R is the circumradius.^{[3]}^{:p. 73}
If are the distances from the vertices of a regular gon to any point on its circumcircle, then ^{[2]}
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.^{[6]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. The list 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 
The area A of a convex regular nsided polygon having side s, circumradius R, apothem a, and perimeter p is given by^{[7]}^{[8]}
For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table:^{[9]} (Note that since as ^{[10]}, 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 circumcirclearea 
Exact  Approximation  As an (approximate) multiple of incirclearea  
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  ^{[11]}  17.64236291  ^{[12]}  3.050524822  0.9710122088  ^{[13]}  3.188348426  1.014882824 
16  ^{[14]}  20.10935797  3.061467460  0.9744953584  ^{[15]}  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  ^{[16]}  31.56875757  ^{[17]}  3.090169944  0.9836316430  ^{[18]}  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.^{[19]}
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,^{[20]}^{:p. xi} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.^{[20]}^{:pp. 4950} 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 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 GaussWantzel theorem.
Equivalently, a regular ngon is constructible if and only if the cosine of its common angle is a constructible numberthat is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.
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.
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) 
^{[21]} 
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:
m and n must be coprime, or the figure will degenerate.
The degenerate regular stars of up to 12 sides are:
Grünbaum {6/2} or 2{3}^{[22]} 
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:
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.
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.
1=
(help)
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(?/4)
journal=
(help)CS1 maint: ref=harv (link)