12gons

A 12gon, also known as a dodecagon, is a polygon with twelve sides and twelve vertices. In Euclidean geometry, a regular dodecagon has twelve equal sides and twelve equal interior angles. The interior angle of a regular dodecagon measures 150 degrees, and the exterior angle measures 30 degrees. The sum of the interior angles of any dodecagon, regular or irregular, is (12-2) * 180 = 1800 degrees.

The area of a regular dodecagon with side length 'a' can be calculated using the formula Area = 3(2 + sqrt(3))a^2. Alternatively, if 'R' is the radius of the circumscribed circle, the area is given by Area = 3R^2. If 'r' is the radius of the inscribed circle (apothem), the area is Area = 12r^2 * tan(15 degrees) or Area = 12r^2 * (2 - sqrt(3)).

Dodecagons appear in various contexts. Architecturally, they can be found in the design of some buildings and floor patterns. In nature, while not as common as polygons with fewer sides, some crystals and cellular structures can exhibit dodecagonal symmetry. The dodecagon is also a fundamental shape in some tiling patterns and geometric constructions. Historically, its properties have been studied in relation to other regular polygons, particularly in the context of constructible polygons.