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.
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.
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.
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.
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. For this reason, a circle is not a polygon with an infinite number of sides.
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.
For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem: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.
The sum of the perpendiculars from a regular n-gon's vertices to any line tangent to the circumcircle equals n times the circumradius.: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.:p.73
The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR2 − ns2/4, where s is the side length and R is the circumradius.:p. 73
Of all n-gons with a given perimeter, the one with the largest area is regular.
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,:p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.:pp. 49–50 This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?
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. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
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.
More generally regular skew polygons can be defined in n-space. 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.
All regular polygons are self-dual to congruency, and for odd n they are self-dual to identity.
In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.
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.
A polyhedron having regular triangles as faces is called a deltahedron.