# Dodecagon

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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:

Or, if *R* is the radius of the circumscribed circle,^{[1]}

And, if *r* is the radius of the 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 dodecagon when it sits on a side as base, and the diameter of the inscribed circle.

The perimeter for an inscribed dodecagon of radius 1 is 12√(2 - √3), or approximately 6.21165708246. ^{[2]}

The perimeter for a circumscribed dodecagon of radius 1 is 24(2 – √3), or approximately 6.43078061835. Interestingly, this is double the value of the area of the *inscribed* dodecagon of radius 1. ^{[3]}

With respect to the above-listed equations for area and perimeter, when the radius of the inscribed dodecagon is 1, note that the area of the inscribed dodecagon is 12(2 – √3) and the *perimeter* of this same inscribed dodecagon is 12√(2 - √3).

## 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

## Dissection

A regular dodecagon can be dissected into a central hexagon, and alternating triangles and squares |
Dodecagon made with pattern blocks |

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.^{[4]}

## 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 |

### Petrie polygons

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

## Examples in use

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

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".

Regular dodecagonal coins include:

- British threepenny bit from 1937 to 1971, when it ceased to be legal tender.
- British One Pound Coin to be introduced in 2017.
- 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

- ↑ See also Kürschák's geometric proof on the Wolfram Demonstration Project
- ↑
*Plane Geometry: Experiment, Classification, Discovery, Application*by Clarence Addison Willis B., (1922) Blakiston's Son & Company, p. 249 [1] - ↑
*Elements of geometry*by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 [2] - ↑ "Doin' Da' Dodeca'" on mathforum.org

## External links

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

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