18gon

An 18-gon, also called an octadecagon, is a polygon with 18 sides and 18 vertices. The term octadecagon derives from Greek roots meaning eighteen sides. A regular 18-gon has all sides equal and all interior angles equal.

In a regular 18-gon, each interior angle measures 160 degrees, and each exterior angle measures 20 degrees (the sum of exterior angles is always 360 degrees). The total sum of interior angles is (n−2)×180 = 2880 degrees. The central angle subtending each side is 360/18 = 20 degrees, and the polygon has dihedral symmetry D18 (18 rotational symmetries and 18 reflections).

The number of diagonals in an 18-gon is n(n−3)/2 = 18×15/2 = 135. If a circumscribed circle is present with radius R, the area of a regular 18-gon can be expressed as A = (1/2) n R^2 sin(2π/n) = 9 R^2 sin(20°). Equivalently, with side length s, A = (1/4) n s^2 cot(π/n) = (9/2) s^2 cot(10°).

Constructibility: a regular 18-gon is not constructible with straightedge and compass, since 18 = 2 × 3^2 does not fit the standard criterion (a form 2^k times a product of distinct Fermat primes).

Regular 18-gons do not tile the plane by themselves, and in general they are studied as a specific case within the family of regular polygons, notable for their 20-degree central angle and 160-degree interior angles.