Dodecagons

A dodecagon is a polygon with twelve sides and twelve interior angles. A regular dodecagon has all sides equal and all interior angles equal, each measuring 150 degrees.

The sum of the interior angles of any dodecagon is 1800 degrees, and its exterior angles total 360 degrees, with each exterior angle measuring 30 degrees in a regular form. The vertices of a regular dodecagon lie on a common circle, and the central angle between adjacent vertices is 30 degrees. If the circumradius is R, the area is A = 3R^2, while the side length s = 2R sin(15°) and the apothem a = R cos(15°). A regular dodecagon has 54 diagonals.

The symmetry of a regular dodecagon is the dihedral group D12, consisting of 12 rotations and 12 reflections, for a total of 24 symmetries. It is constructible with straightedge and compass because 12 = 2^2 × 3, a product of a power of two and a Fermat prime.

In tiling and design, dodecagons appear in the uniform tiling 4.6.12, where each vertex is surrounded by a square, a hexagon, and a dodecagon. They also occur in decorative patterns and in modern currency, including the 12-sided British pound coin.

The name derives from Greek dōdeka, "twelve," and gōnía, "angle."