Dodecagon
From Wikipedia
| Regular dodecagon | |
|---|---|
 A regular dodecagon |
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| Type | Regular polygon |
| Edges and vertices | 12 |
| Schläfli symbol | {12} t{6} |
| Coxeter–Dynkin diagram |     |
| Symmetry group | Dihedral (D12) |
| Internal angle (degrees) | 150° |
| Properties | convex, cyclic, equilateral, isogonal, isotoxal |
In geometry, a dodecagon is any polygon with twelve sides and twelve angles.
Table of Contents
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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 the circumscribed circle,[1]
And, if r is the radius of the inscribed circle,
A simple formula for area (given the two measurements) is: View formula on Wikipedia 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, View formula on Wikipedia.
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:
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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
One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.[2]
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.
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.
Notes
- ↑ See also Kürschák's geometric proof on the Wolfram Demonstration Project
- ↑ "Doin' Da' Dodeca'" on mathforum.org
External links
- Weisstein, Eric W., "Dodecagon" from MathWorld.
- Kürschak's Tile and Theorem
- Definition and properties of a dodecagon With interactive animation
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