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My name is Madelaine and I am studying Industrial and Labor Relations and Continuing Education and Summer Sessions at Kleindorf Ii / Austria.

my web site: fitness exercise programs In geometry, a dodecagon is any polygon with twelve sides and twelve angles.

Regular dodecagon

A regular dodecagon has all sides of equal length and all angles equal to 150°. It has 12 lines of symmetry and rotational symmetry of order 12. Its Schläfli symbol is {12}.

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

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

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,

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

A simple formula for area (given the two measurements) is: $A=3ad$ 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, $d=a(1+2cos30^{\circ }+2cos60^{\circ })$.

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	File:11-simplex t1.svg Rectified 11-simplex	File:11-simplex t2.svg Birectified 11-simplex	File:11-simplex t3.svg Trirectified 11-simplex	File:11-simplex t4.svg Quadrirectified 11-simplex	File:11-simplex t5.svg Quintirectified 11-simplex
BC6	File:6-cube t5.svg 6-orthoplex	File:6-cube t4.svg Rectified 6-orthoplex	File:6-cube t3.svg Birectified 6-orthoplex	File:6-cube t2.svg Birectified 6-cube	File:6-cube t1.svg Rectified 6-cube	File:6-cube t0.svg 6-cube
D7	File:7-cube t6 B6.svg t5(141)	File:7-cube t5 B6.svg t4(141)	File:7-cube t4 B6.svg t3(141)	File:7-cube t3 B6.svg t2(141)	File:7-demicube t1 D7.svg t1(141)	File:7-demicube t0 D7.svg t0(141)
E6	File:E6 graph.svg t0(221)	File:Up 2 21 t1 E6.svg t1(221)	File:Up 2 21 t2 E6.svg t1(122)	File:Gosset 1 22 polytope.png t0(122)
F4	File:24-cell t0 F4.svg 24-cell	File:24-cell t1 F4.svg Rectified 24-cell	File:24-cell h01 F4.svg Snub 24-cell

Examples in use

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

File:Segovia Vera Cruz.jpg

The Vera Cruz church in Segovia

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".

File:1942 threepence reverse.jpg

A 1942 British threepence, reverse

Regular dodecagonal coins include:

• British threepenny bit from 1937 to 1971, when 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.
• Dodecagram

Notes

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• Kürschak's Tile and Theorem
• Definition and properties of a dodecagon With interactive animation

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