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: The symmetry group of an n-sided regular polygon is the dihedral group Dn (of order 2n): D2, D3, D4, ...

It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center.

If n is odd then all axes pass through a vertex and the midpoint of the opposite side.

An n-sided convex regular polygon is denoted by its Schläfli symbol

, we have two degenerate cases: In certain contexts all the polygons considered will be regular.

For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°.

As the number of sides increases, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle.

The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line (see apeirogon).

For this reason, a circle is not a polygon with an infinite number of sides.

is the distance from an arbitrary point in the plane to the centroid of a regular

For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem[4]: p. 72  (the apothem being the distance from the center to any side).

The sum of the perpendiculars from a regular n-gon's vertices to any line tangent to the circumcircle equals n times the circumradius.[4]: p.

73 The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR2 where R is the circumradius.[4]: p.

-gon to any point on its circumcircle, then [3] Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into

These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m-cubes.

[7] In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi.

The list OEIS: A006245 gives the number of solutions for smaller polygons.

The area A of a convex regular n-sided polygon having side s, circumradius R, apothem a, and perimeter p is given by[8][9]

For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table:[b] (Since

49–50  This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge?

Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796.

Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.

Equivalently, a regular n-gon 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.

A regular skew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism.

More generally regular skew polygons can be defined in n-space.

The (non-degenerate) 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: Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure.

The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids.

A polyhedron having regular triangles as faces is called a deltahedron.