Dodecagon

And in terms of the apothem r (see also inscribed figure), the area is:

The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is:

This coefficient is double the coefficient found in the apothem equation for area.[3]

As 12 = 22 × 3, regular dodecagon is constructible using compass-and-straightedge construction:

Construction of a regular dodecagon
at a given side length, animation. (The construction is very similar to that of octagon at a given side length.)

Coxeter states that every zonohedron (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular dodecagon, m=6, and it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs. This decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. The sequence OEIS sequence A006245 defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection.

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.[5] They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.

The regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges.

A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways:

Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration:

A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism.

A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes.

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. In 6 dimensions 6-cube, 6-orthoplex, 221, 122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A cross is a dodecagon, as is the logo for the Chevrolet automobile division.

The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military watchtower in Seville, southern Spain, built by the Almohad dynasty. The early thirteenth century Vera Cruz church in Segovia, Spain is dodecagonal. Another example is the Porta di Venere (Venus' Gate), in Spello, Italy, built in the 1st century BC has two dodecagonal towers, called "Propertius' Towers".

In the Philippines, in local carnivals (peryahan), ferris wheels usually with 12 seats or gondolas