# Dodecagon

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

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. \end{align}

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. \end{align}

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).

## 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 Regular 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

## Related figures

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
{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 (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}.[5]

Vertex-transitive truncations of the hexagon
Quasiregular Isogonal Quasiregular

t{6}={12}

t{6/5}={12/5}

### Petrie polygons

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

E6 F4 2G2 (4D)

221

122

24-cell

Snub 24-cell

6-6 duopyramid

6-6 duoprism
A11 D7 B6

11-simplex

(411)

141

6-orthoplex

6-cube

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

## Examples in use

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: