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,

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

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 dodecahedron when it sits on a side as base, and the diameter of the inscribed circle.

By simple trigonometry, $\scriptstyle d\,=\,a(1\,+\,2cos{30^\circ}\,+\,2cos{60^\circ})$.

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

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

Pattern blocks

Dodecagon made with pattern blocks

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

Petrie polygons

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

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

• British threepenny bit from 1937 to 1971, at which time it ceased to be legal tender.
• 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

• Dodecagonal number
• Dodecahedron – a regular polyhedron with 12 pentagonal faces.
• Dodecagram

External links

• Weisstein, Eric W., "Dodecagon", MathWorld.
• Kürschak's Tile and Theorem
• Definition and properties of a dodecagon With interactive animation