A regular dodecagon
|Edges and vertices||12|
|Symmetry group||D12, order 2×12|
|Internal angle (degrees)||150°|
|Properties||convex, cyclic, equilateral, isogonal, isotoxal|
The area of a regular dodecagon with side a is given by:
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.
By simple trigonometry, .
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).
A regular dodecagon can fill a plane vertex with other regular polygons:
Here are 3 example periodic plane tilings that use dodecagons:
Semiregular tiling 3.12.12
Semiregular tiling: 4.6.12
A demiregular tiling:
126.96.36.199 & 188.8.131.52.3.3
Examples in use
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 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 
- Elements of geometry by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 
- "Doin' Da' Dodeca'" on mathforum.org
- Weisstein, Eric W., "Dodecagon", MathWorld.
- Kürschak's Tile and Theorem
- Definition and properties of a dodecagon With interactive animation