dodekagon

A dodekagon, also called a dodecagon, is a polygon with twelve sides and twelve interior angles. The name derives from the Greek dodeka meaning twelve and gónía meaning angle. While dodecagon is the common English form, dodekagon is a variant spelling used in some languages.

In the regular dodekagon, all sides and all interior angles are equal. The figure has twelve sides, twelve vertices, and each interior angle measures 150 degrees (the exterior angle is 30 degrees). The polygon can be inscribed in or circumscribed about a circle, with the central angle corresponding to each side equal to 30 degrees. The regular dodecagon has dihedral symmetry of order 24 (the symmetry group D12), consisting of 12 reflections and 12 rotations.

Formulas and properties: If the dodecagon is inscribed in a circle of radius R, its area is A = 3R^2. If each side has length s, the area is A = 3s^2 cot(π/12) ≈ 11.196s^2. The perimeter is P = 12s. The interior angle sum for any dodecagon is 1800 degrees.

Occurrence and context: Dodecagons appear in geometry problems, design, and decorative patterns. They are not among the regular tilings of the plane by a single polygon, but can be combined with other shapes in semi-regular tilings. The term emphasizes twelve-sided geometry and is related to other polygon names formed from Greek numerical prefixes.