Dodecagon

Template:Even polygon stat table 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:

${\begin{aligned}A&=3\cot \left({\frac {\pi }{12}}\right)a^{2}=3\left(2+{\sqrt {3}}\right)a^{2}\\&\simeq 11.19615242\,a^{2}.\end{aligned}}$

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

$A=6\sin \left({\frac {\pi }{6}}\right)R^{2}=3R^{2}.$

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

${\begin{aligned}A&=12\tan \left({\frac {\pi }{12}}\right)r^{2}=12\left(2-{\sqrt {3}}\right)r^{2}\\&\simeq 3.2153903\,r^{2}.\end{aligned}}$

A simple formula for area (given the two measurements) is: $\scriptstyle A\,=\,3ad$ 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, $\scriptstyle d\,=\,a(1\,+\,2cos{30^{\circ }}\,+\,2cos{60^{\circ }})$.

Uses

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

3.12.12

4.6.12

3.3.4.12

3.4.3.12

Dodecagon construction

A regular dodecagon is constructible using compass and straightedge:

Construction of a regular dodecagon

Occurrence

Tiling

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

Pattern blocks

Dodecagon made with pattern blocks

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.

Petrie polygons

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

A11	11-simplex	Rectified 11-simplex	Birectified 11-simplex	Trirectified 11-simplex	Quadrirectified 11-simplex	Quintirectified 11-simplex
BC6	6-orthoplex	Rectified 6-orthoplex	Birectified 6-orthoplex	Birectified 6-cube	Rectified 6-cube	6-cube
D7	t5(141)	t4(141)	t3(141)	t2(141)	t1(141)	t0(141)
E6	t0(221)	t1(221)	t1(122)	t0(122)
F4	24-cell	Rectified 24-cell	Snub 24-cell

Examples in use

A 1942 British threepence, reverse

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines.

Regular dodecagonal coins include:

• British threepenny bit from 1937 to 1971, at which time it ceased to be legal tender.
• Australian 50-cent coin
• Fijian 50 cents
• Tongan 50-seniti, since 1974
• Solomon Islands 50 cents
• Croatian 25 kuna
• Romanian 5000 lei, 2001–2005
• Canadian penny, 1982–1996
• South Vietnamese 25 đồng, 1968–1975
• Zambian 50 ngwee, 1969–1992
• Malawian 50 tambala, 1986–1995
• Mexican 20 centavos, since 1992

See also

• Dodecagonal number
• Dodecahedron – a regular polyhedron with 12 pentagonal faces.

External links

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