Octadecagon (18-sided polygon)

Properties

The regular octadecagon has a number of elementary metric and symmetry facts that follow from n-gon formulae. Each interior angle measures 160°, and the exterior (central) angle is 20°. Its full dihedral symmetry group is D18, of order 36, containing rotations and reflections.

• Interior angle: (n-2)·180°/n = 160° for n=18.
• Central angle: 360°/18 = 20°.
• Area (regular): A = (n/4)s^2 cot(π/n) = (9/2)s^2 cot(10°), where s is the side length.
• Alternative area: A = (1/2)nR^2 sin(2π/n) = 9R^2 sin(20°), in terms of circumradius R.

There are also classic star polygon forms with Schläfli symbols such as {18/5}, {18/7}, and others, produced by connecting every k-th vertex of a regular 18-gon. These produce a variety of intersecting figures with their own symmetry.

Construction and history

Polygonal geometry was studied by classical mathematicians, and names come from Greek roots meaning "eighteen angles." The regular octadecagon is not constructible with straightedge and compass alone because 18 factors as 2·3^2 and includes the square of an odd prime factor; Gauss–Wantzel criteria require the odd prime factors to be distinct Fermat primes. Practical approximations or constructions with angle trisection or added tools can produce a precise 18-gon.

In practice octadecagons appear in decorative design, tessellation experiments, and as theoretical examples in symmetry and polygonal decomposition. Their moderate number of sides makes them useful in illustrating polygon formulae and relationships between side length, radius, and area in elementary geometry.