Dodecagon

Failed to parse (Missing texvc executable; please see math/README to configure.):  = 3 \left( 2+\sqrt{3} \right) t^2
Failed to parse (Missing texvc executable; please see math/README to configure.):  \simeq 11.19615242 t^2.

In geometry, a dodecagon is any polygon with twelve sides and twelve angles.

Regular dodecagon

It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°. Its Schläfli symbol is {12}.

The area of a regular dodecagon with side a is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 3 \cot\left( \frac{\pi}{12} \right) a^2 = 3 \left( 2+\sqrt{3} \right) a^2 \simeq 11.19615242 a^2.

Or, if R is the radius of a circumscribed circle,

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 6 \sin\left( \frac{\pi}{6}\right) R^2 = 3 R^2.

And, if r is the radius of a inscribed circle,

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 12 \tan\left( \frac{\pi}{12}\right) r^2 = 12 \left( 2-\sqrt{3} \right) r^2 \simeq 3.2153903 r^2.

Dodecagon construction

A regular dodecagon is constructible with compass and straightedge. The following is a 23 step animation illustrating one way it can be done. Notice that the compass radius is unaltered during steps 8 through 11.

Tilings

Here are 3 example periodic plane tilings that uses dodecagons:

Semiregular tiling 3.12.12

Semiregular tiling: 4.6.12

A demiregular tiling: 3.3.4.12 & 3.3.3.3.3.3

Examples in use

• When spelled uppercase, the outlines of the letters E and H (and I in a slab serif font) are all dodecagons.
• The Australian 50-cent coin, Fijian 50-cent, Tongan 50-seniti, and Solomon Island 50-cent coin are regular dodecagons. Croatian 25 kuna coin has a dodecagonal shape as well. Until July 2005, a Romanian coin (5000 ROL) was also dodecagonal. The Canadian penny was dodecagonal from 1982 to 1996 as well as the South Vietnamese 20 Ðong coin until 1975. Zambian 50 ngwee (until 1992) and Malawian 50 tambala (until 1995) coins were dodecagonal. The Mexican 20 cent coin is also 12-sided.

See also

• dodecagonal number
• Dodecahedron - a regular polyhedron with 12 pentagonal faces.

External links

• Dodecagon and Kurschak's Tile and Theorem by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Eric W. Weisstein, Dodecagon at MathWorld.
• Kürschak's Tile and Theorem
• Definition and properties of a dodecagon With interactive animation

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