Dodecagon

A dodecagon is a plane figure (polygon) consisting of twelve straight sides and twelve vertices. It belongs to the category of polygons with many sides, and is known in other languages as a dodecagon. The Somali name follows the same formation pattern used for the triangle (saddexaale) and quadrilateral (afaraale): the number of sides with a suffix added.

There are two main types: the regular dodecagon, in which all sides are equal and all angles are equal, and the irregular dodecagon, which has sides and angles of different measures. They may also be classified as convex, with no interior angles pointing inward, or concave.

Angle properties

The sum of the interior angles of any twelve-sided polygon is 1800 degrees, following the formula (n − 2) × 180, where n is the number of sides. In a regular dodecagon, each interior angle measures 150 degrees, and each exterior angle measures 30 degrees. The sum of the exterior angles is always 360 degrees.

Interior diagonals and symmetry

The number of interior diagonals (diagonals) is calculated as n(n − 3) / 2, giving a dodecagon 54 diagonals. The regular dodecagon has twelve lines of symmetry and rotational symmetry of order 12, meaning it returns to its original position when rotated by 30 degrees.

Area and perimeter

If each side has length a, the area of a regular dodecagon is given by the formula 3(2 + √3)a², approximately equal to 11.196a². The perimeter is 12a. A regular dodecagon can be inscribed in a circle with all vertices touching it, making it a cyclic polygon.

Applications

This shape appears in architectural and geometric decoration, particularly in Islamic art based on circle division. It is also used in the design of certain coins and technical instruments, since the many sides approximate a circle while still maintaining distinct edges for handling.