Dodecagon


Dodecagon
Regular dodecagon
Regular polygon 12.svg
Edges and vertices 12
Schläfli symbols {12}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 12.pngCDel node.png
Symmetry group Dihedral (D12)
Internal angle
(degrees)
150°

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

Contents

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:

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

Occurrence

Here are 3 example periodic plane tilings that use dodecagons:

Tile 3bb.svg
Semiregular tiling 3.12.12
Tile 46b.svg
Semiregular tiling: 4.6.12
Dem3343tbc.gif
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 t0.svg
11-simplex
11-simplex t1.svg
Rectified 11-simplex
11-simplex t2.svg
Birectified 11-simplex
11-simplex t3.svg
Trirectified 11-simplex
11-simplex t4.svg
Quadrirectified 11-simplex
11-simplex t5.svg
Quintirectified 11-simplex
BC6 6-cube t5.svg
6-orthoplex
6-cube t4.svg
Rectified 6-orthoplex
6-cube t3.svg
Birectified 6-orthoplex
6-cube t2.svg
Birectified 6-cube
6-cube t1.svg
Rectified 6-cube
6-cube t0.svg
6-cube
D7 7-cube t6 B6.svg
t5(141)
7-cube t5 B6.svg
t4(141)
7-cube t4 B6.svg
t3(141)
7-cube t3 B6.svg
t2(141)
7-demicube t1 D7.svg
t1(141)
7-demicube t0 D7.svg
t0(141)
E6 E6 graph.svg
t0(221)
Up 2 21 t1 E6.svg
t1(221)
Up 2 21 t2 E6.svg
t1(122)
Gosset 1 22 polytope.png
t0(122)
F4 24-cell t0 F4.svg
24-cell
24-cell t1 F4.svg
Rectified 24-cell
24-cell h01 F4.svg
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:

See also

External links


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Look at other dictionaries:

  • dodecagon — DODECAGÓN, dodecagoane, s.n. Poligon cu douăsprezece laturi (latură) şi douăsprezece unghiuri. – Din fr. dodécagone. Trimis de claudia, 13.09.2007. Sursa: DEX 98  dodecagón s. n., pl. dodecagoáne Trimis de siveco, 10.08.2004. Sursa: Dicţionar… …   Dicționar Român

  • Dodecagon — Do*dec a*gon, n. [Gr. ? twelve + ? angle: cf. F. dod[ e]cagone.] (Geom.) A figure or polygon bounded by twelve sides and containing twelve angles. [1913 Webster] || …   The Collaborative International Dictionary of English

  • dodecagon — ► NOUN ▪ a plane figure with twelve straight sides and angles …   English terms dictionary

  • dodecagon — [dō dek′ə gän΄, dō dek′əgən] n. [Gr dōdekagōnon: see DODECA & GON] a plane figure with twelve angles and twelve sides …   English World dictionary

  • dodecagon —    A closed shape bounded by twelve straight line segments. The formula with which to find an equilateral dodecagon s area is 11.1961 times the length of one side squared. Also see circle, dodecahedron, mathematics, polygon, radial, shape, and… …   Glossary of Art Terms

  • dodecagon — noun Etymology: Greek dōdekagōnon, from dōdeka + gōnon gon Date: circa 1658 a polygon of 12 angles and 12 sides …   New Collegiate Dictionary

  • dodecagon — dodecagonal /doh de kag euh nl/, adj. /doh dek euh gon , geuhn/, n. Geom. a polygon having 12 angles and 12 sides. Also, duodecagon. [1650 60; DODECA + GON] * * * …   Universalium

  • dodecagon — noun a polygon with twelve edges and twelve angles Syn: duodecagon See Also: dodecagonal …   Wiktionary

  • dodecàgon — do|de|cà|gon Mot Pla Nom masculí dodecasíl·lab do|de|ca|síl|·lab Mot Pla Adjectiu variable dodecasíl·lab do|de|ca|síl|·lab Mot Pla Nom masculí …   Diccionari Català-Català

  • dodecagón — s. n., pl. dodecagoáne …   Romanian orthography