dodecagon

Regular dodecagon
Edges and vertices	12
Schläfli symbols	{12}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D12)
Area (with t=edge length)
Internal angle (degrees)	150°

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:

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

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

A simple formula for area (given the two measurements) is:

where d is the distance between parallel sides.

Length d is the height of the dodecahedron when it sits on a side as base, and the diameter of the inscribed circle.

By simple trigonometry, d = a(1 + 2cos30° + 2cos60°).

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.

Occurrence

Here are 3 example periodic plane tilings that use dodecagons:

Semiregular tiling 3.12.12

Semiregular tiling: 4.6.12

A demiregular tiling: 3.3.4.12 & 3.3.3.3.3.3

Petrie polygons

The regular dodecagon is the Petrie polygon for eight higher dimensional polytopes, shown in these skew orthogonal projections:

24-cell (4D)	6-cube (6D)	6-orthoplex (6D)	Rectified 6-orthoplex (6D)
E6 polytope (6D)	122 (6D)	7-demicube (7D)	11-simplex (11D)

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 British threepenny bit was a regular dodecagon from 1937 to 1971, at which time it ceased to be legal tender. 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.
• The E6 polytope can be drawn with an orthogonal projection inside a regular dodecagon.

External links

• Weisstein, Eric W., "Dodecagon" from MathWorld.
• Kürschak's Tile and Theorem
• Definition and properties of a dodecagon With interactive animation