Regularpolygon

Regularpolygon

In geometry, a regular polygon, or regular n-gon, is a polygon with all sides equal in length and all interior angles equal. This means the polygon is both equilateral and equiangular. Regular polygons are typically assumed to be simple and convex, though non-convex regular star polygons can be considered in extended definitions. The number of sides, n, is an integer at least 3.

Key properties include that all vertices lie on a common circle (the circumcircle), and the center is equidistant from all vertices (the circumradius R). The distance from the center to each side is the inradius r. The central angle subtended by a side is 360°/n, and the interior angle is α = (n−2)×180°/n.

Formulas: If the side length is s, the area is A = n s^2 / (4 tan(π/n)). Equivalently, with circumradius R, A = (1/2) n R^2 sin(2π/n). The side length relates to R by s = 2 R sin(π/n), and the inradius by r = R cos(π/n).

Regular polygons and tiling: Only triangles (n = 3), squares (n = 4), and hexagons (n = 6) tile the plane by themselves. Other regular polygons do not tessellate the plane without gaps. In symmetry terms, a regular n-gon has dihedral symmetry D_n.

Extensions: regular star polygons, denoted {n/m} with gcd(n,m) = 1 and 1 ≤ m < n/2, generalize the concept to certain non-convex shapes.