dodecagon

A dodecagon is a polygon with twelve sides and twelve vertices. When all sides and all interior angles are equal, it is called a regular dodecagon; if not, the shape is simply a dodecagon or an irregular dodecagon.

In a regular dodecagon, the central angle subtended by each side is 360/12 = 30 degrees, and the interior angle at each vertex is (12−2)×180/12 = 150 degrees. The symmetry group is the dihedral group D12, consisting of 12 rotations and 12 reflections. The maximum number of diagonals in a dodecagon is n(n−3)/2 = 12×9/2 = 54.

For a regular dodecagon with side length s, the perimeter is P = 12s, and the area is A = (12 s^2)/(4 tan(π/12)) = 3 s^2 (2+√3) ≈ 11.196 s^2. If the circumradius R (distance from the center to a vertex) is used, then A = 3R^2 and the side length satisfies s = 2R sin(π/12). The apothem (the inradius) a relates to the area by A = (1/2) P a.

A convenient coordinate model places a regular dodecagon centered at the origin with vertices at (R cos(kπ/6), R sin(kπ/6)) for k = 0,1,...,11.

Dodecagons can appear in plane tilings with other regular polygons; for example, the Archimedean tiling with vertex configuration 4.6.12 uses squares, hexagons, and dodecagons meeting at each vertex.