In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. 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.
General properties
These properties apply to all regular polygons, whether convex or star.
A regular n-sided 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 equal-length 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 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.
A regular n-sided polygon can be constructed with origami if and only if for some , where each distinct is a Pierpont prime.
Symmetry
The symmetry group of an n-sided regular polygon is 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 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 n-sided 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 n-gon, each interior angle has a measure of: 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 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 (see apeirogon). 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 .
For a regular n-gon inscribed in a unit-radius 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 n-gon with circumradius R and distances di from an arbitrary point in the plane to the vertices, we have
For higher powers of distances from an arbitrary point in the plane to the vertices of a regular -gon, if
,
then
,
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
, where = 1, 2, …, .