Dodecagon

A regular dodecagon is represented by the Schläfli symbol {12} and can be constructed as a truncated hexagon, t{6}, or a twice-truncated triangle, tt{3}.

A simple formula for area (given side length and span) is: This can be verified with the trigonometric relationship: The perimeter of a regular dodecagon in terms of circumradius is:[2] The perimeter in terms of apothem is: This coefficient is double the coefficient found in the apothem equation for area.

[5] They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.

In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, [2+,10] symmetry, order 20.

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes.

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths.

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.

Three squares of sides R can be cut and rearranged into a dodecagon of circumradius R , yielding a proof without words that its area is 3 R 2
Isotoxal dodecagon
The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal, diasymmetry) with mirror lines through vertices, p with mirror lines through edges (perpendicular, persymmetry) i with mirror lines through both vertices and edges (isosymmetry), and g for rotational (gyrosymmetry). a1 labels asymmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons. [ 6 ]
A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism .
The Vera Cruz church in Segovia
A 1942 British threepence, reverse