Dodecagon
Regular dodecagon  

A regular dodecagon  
Type  Regular polygon 
Edges and vertices  12 
Schläfli symbol  {12}, t{6}, tt{3} 
Coxeter diagram 

Symmetry group  Dihedral (D_{12}), order 2×12 
Internal angle (degrees)  150° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a dodecagon or 12gon is any twelvesided polygon.
Regular dodecagon
A regular dodecagon is a closed figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol {12} and can be constructed as a truncated hexagon, t{6}, or a twicetruncated triangle, tt{3}. The internal angle at each vertex of a regular dodecagon is 150°.
Area
The area of a regular dodecagon of side length a is given by:
And in terms of the apothem r (see also inscribed figure), the area is:
In terms of the circumradius R, the area is:^{[1]}
The span of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is:
This can be verified with the trigonometric relationship:
Perimeter
The perimeter of a regular dodecagon in terms of apothem is:^{[2]}
The perimeter in terms of circumradius is:
This coefficient is double the coefficient found in the apothem equation for area.^{[3]}
Dodecagon construction
As 12 = 2^{2} × 3, regular dodecagon is constructible using compass and straightedge:
Dissection
Coxeter states that every parallelsided 2mgon can be divided into m(m1)/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs, with one example shown below. This decomposition is based on a Petrie polygon projection of a 6cube, with 15 of 240 faces.^{[4]}
With hexagons, squares, and triangles 
pattern blocks 
With 15 rhombs from 6cube 
With 15 rhombs 
One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.^{[5]}
Symmetry
The regular dodecagon has Dih_{12} symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges.
Example dodecagons by symmetry  

r24  
d12 
g12 
p12 
i8  
d6 
g6 
p6 
d4 
g4 
p4  
g3 
d2 
g2 
p2  
a1 
Occurrence
Tiling
A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways:
3.12.12  4.6.12  3.3.4.12  3.4.3.12 

Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration:
1uniform  2uniform  

3.12.12 
4.6.12 
3.12.12; 3.4.3.12 
Skew dodecagon
A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such an dodecagon is not generally defined. A skew zigzag dodecagon has vertices alternating between two parallel planes.
A regular skew dodecagon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D_{5d}, [2^{+},10] symmetry, order 20. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossedantiprism, s{2,24/7} also have regular skew dodecagons.
Petrie polygons
The regular dodecagon is the Petrie polygon for many higherdimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24cell, snub 24cell, 66 duoprism, 66 duopyramid. In 6 dimensions 6cube, 6orthoplex, 2_{21}, 1_{22}. It is also the Petrie polygon for the grand 120cell and great stellated 120cell.
Regular skew dodecagons in higher dimensions  

E_{6}  F_{4}  2G_{2} (4D)  
2_{21} 
1_{22} 
24cell 
Snub 24cell 
66 duopyramid 
66 duoprism 
A_{11}  D_{7}  B_{6}  
11simplex 
(4_{11}) 
1_{41} 
6orthoplex 
6cube 
Related figures
A dodecagram is a 12sided star polygon, represented by symbol {12/n}. There is one regular star polygon: {12/5}, using the same vertices, but connecting every fifth point. There are also three compounds: {12/2} is reduced to 2{6} as two hexagons, and {12/3} is reduced to 3{4} as three squares, {12/4} is reduced to 4{3} as four triangles, and {12/6} is reduced to 6{2} as six degenerate digons.
Stars and compounds  

n  1  2  3  4  5  6 
Form  Polygon  Compounds  Star polygon  Compound  
Image  {12/1} = {12} 
{12/2} or 2{6} 
{12/3} or 3{4} 
{12/4} or 4{3} 
{12/5} 
{12/6} or 6{2} 
Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertextransitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t{6}={12}. A quasitruncated hexagon, inverted as {6/5}, is a dodecagram: t{6/5}={12/5}.^{[7]}
Vertextransitive truncations of the hexagon  

Quasiregular  Isogonal  Quasiregular  
t{6}={12} 
t{6/5}={12/5} 
Examples in use
In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A cross is a dodecagon.
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 50cent coin
 Fijian 50 cents
 Tongan 50seniti, 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, 19922009
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
 ↑ Elements of geometry by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ "Doin' Da' Dodeca'" on mathforum.org
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
External links
 Kürschak's Tile and Theorem
 Definition and properties of a dodecagon With interactive animation