Dodecagon

Regular dodecagon
Edges and vertices	12
Schläfli symbols	{12}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D12)
Internal angle (degrees)	150°

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

Regular dodecagon

It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°. 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:

$\ A = 3ad$ 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, d = a(1 + 2cos30° + 2cos60°).

Dodecagon construction

A regular dodecagon is constructible using compass and straightedge:

Construction of a regular dodecagon

Occurrence

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	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 and H (and I in a slab serif font) have dodecagonal outlines.

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.

External links

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