From Wikipedia, the free encyclopedia
Jump to: navigation, search
Regular dodecagon
Regular polygon 12 annotated.svg
A regular dodecagon
Type Regular polygon
Edges and vertices 12
Schläfli symbol {12}
Coxeter diagram CDel node 1.pngCDel 12.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.png
Symmetry group Dihedral (D12), order 2×12
Internal angle (degrees) 150°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a dodecagon is any polygon with twelve sides and twelve angles.

A regular dodecagon has Schläfli symbol {12} and can be constructed as a quasiregular truncated hexagon, t{6}, which alternates two types of edges.

Regular dodecagon[edit]

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{align} A & = 3 \cot\left(\frac{\pi}{12} \right) a^2 = 
                     3 \left(2+\sqrt{3} \right) a^2 \\
                 & \simeq 11.19615242\,a^2.

Or, if R is the radius of the circumscribed circle,[1]

A = 6 \sin\left(\frac{\pi}{6}\right) R^2 = 3 R^2.

And, if r is the radius of the inscribed circle,

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

A simple formula for area (given the two measurements) is: \scriptstyle A\,=\,3ad 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, \scriptstyle d\,=\,a(1\,+\,2cos{30^\circ}\,+\,2cos{60^\circ}).

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).

Dodecagon construction[edit]

A regular dodecagon is constructible using compass and straightedge:

Regular Dodecagon Inscribed in a Circle.gif
Construction of a regular dodecagon


Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/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 6-cube, with 15 of 240 faces.[4]

Regular dodecagons dissected
Hexagonal cupola flat.png
With hexagons, squares, and triangles
Wooden pattern blocks dodecagon.JPG
pattern blocks
Rhomb dissected dodecagon.png
With 15 rhombs from 6-cube
Rhomb dissected dodecagon3.png
With 15 rhombs

One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons.[5]


The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons. [6]

The regular dodecahedron has Dih12 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
Full symmetry dodecagon.png
Hexagonal star dodecagon.png
Gyrated dodecagon.png
Truncated hexagon dodecagon.png
Cross dodecagon.png
Hexagonal star d6 dodecagon.png
Twisted hexagonal star dodecagon.png
Truncated triangular star dodecagon.png
D4 star dodecagon.png
Twisted cross dodecagon.png
Twisted triangle star dodecagon.png
D2 star dodecagon.png
Distorted twisted cross dodecagon.png
Distorted H-shape-dodecagon.png
No symmetry dodecagon.png



A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways:

Vertex type 3-12-12.svg Vertex type 4-6-12.svg Vertex type 3-3-4-12.svg Vertex type 3-4-3-12.svg
3.12.12 4.6.12

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

1-uniform 2-uniform
Tile 3bb.svg
1-uniform n3.svg
2-uniform n2.svg

Related figures[edit]

A dodecagram is a 12-sided 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.

n 1 2 3 4 5 6
Form Polygon Compounds Star polygon Compound
Image Regular polygon 12.svg
{12/1} = {12}
Regular star figure 2(6,1).svg
{12/2} or 2{6}
Regular star figure 3(4,1).svg
{12/3} or 3{4}
Regular star figure 4(3,1).svg
{12/4} or 4{3}
Regular star polygon 12-5.svg
Regular star figure 6(2,1).svg
{12/6} or 6{2}

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) 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]

Vertex-transitive truncations of the hexagon
Quasiregular Isogonal Quasiregular
Regular polygon truncation 6 1.svg
Regular polygon truncation 6 2.svg Regular polygon truncation 6 3.svg Regular polygon truncation 6 4.svg

Petrie polygons[edit]

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

E6 F4 2G2 (4D)
E6 graph.svg
Gosset 1 22 polytope.png
24-cell t0 F4.svg
24-cell h01 F4.svg
Snub 24-cell
6-6 duopyramid ortho-3.png
6-6 duopyramid
6-6 duoprism ortho-3.png
6-6 duoprism
A11 D7 B6
11-simplex t0.svg
7-cube t6 B6.svg
7-demicube t0 D7.svg
6-cube t5.svg
6-cube t0.svg

It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.

Examples in use[edit]

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

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".

A 1942 British threepence, reverse

Regular dodecagonal coins include:

See also[edit]


  1. ^ See also Kürschák's geometric proof on the Wolfram Demonstration Project
  2. ^ Plane Geometry: Experiment, Classification, Discovery, Application by Clarence Addison Willis B., (1922) Blakiston's Son & Company, p. 249 [1]
  3. ^ Elements of geometry by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 [2]
  4. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  5. ^ "Doin' Da' Dodeca'" on
  6. ^ The Symmetries of Things, Chapter 20
  7. ^ 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[edit]