# Dodecagon

Regular dodecagon
A regular dodecagon
Type Regular polygon
Edges and vertices 12
Schläfli symbol {12}
Coxeter diagram
Symmetry group 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.

## 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 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

### Petrie polygons

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

 A11 BC6 D7 E6 F4 11-simplex Rectified 11-simplex Birectified 11-simplex Trirectified 11-simplex Quadrirectified 11-simplex Quintirectified 11-simplex 6-orthoplex Rectified 6-orthoplex Birectified 6-orthoplex Birectified 6-cube Rectified 6-cube 6-cube t5(141) t4(141) t3(141) t2(141) t1(141) t0(141) t0(221) t1(221) t1(122) t0(122) 24-cell Rectified 24-cell Snub 24-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: