Dodecagon (12-sided polygon)

Basic properties

In a regular dodecagon each interior angle measures 150° and each exterior angle is 30°. A general formula for an n‑gon's interior angle, (n−2)×180°/n, yields 150° when n=12. The number of distinct diagonals of any 12‑gon is 12(12−3)/2 = 54.

Area and lengths

Several useful area and side relationships exist for the regular dodecagon. If R denotes the circumradius (distance from center to a vertex), the area is simple: A = 3R². In terms of side length s, the area can be written A = 3 s² cot(π/12), or more generally A = (1/4) n s² cot(π/n) with n=12. The side length relates to the circumradius by s = 2R sin(π/12).

Symmetry and construction

The regular dodecagon has dihedral symmetry D12 of order 24: twelve rotations (including the identity) and twelve reflections. It is constructible with straightedge and compass because 12 factors as 3×2², a product of a Fermat prime and a power of two; practically, one way to construct it is to mark off twelve equal central angles of 30° on a circle.

Uses, examples and tiling

Dodecagonal outlines appear in architecture, decorative art and coin design; some medallions and coins use twelve‑sided plans for aesthetic or anti‑counterfeiting reasons. A regular dodecagon cannot tile the plane by itself since 360°/150° is not an integer, but it can participate in composite tilings together with other polygons.

Related forms

Beyond the convex regular 12‑gon, one can form star figures by connecting every k‑th vertex of a regular dodecagon; these are denoted {12,k} and produce various dodecagrams. Irregular dodecagons allow many shapes with differing side lengths and angles, useful in modeling and design where twelvefold vertex count is required.