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Dodecagon
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a dodecagon is any polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 with twelve
12 (number)

12 is the natural number following 11 and preceding 13 .The word "twelve" is a native English language word that presumably arises from the Germanic languages compound twa-lif "two-leave", meaning that two is...
 sides and twelve angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s.

if R is the radius of a circumscribed circle,

And, if r is the radius of a inscribe
Inscribe

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Specifically, at all points where figures meet, their edges must lie tangent....
d circle,
gular dodecagon is constructible with compass and straightedge
Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization ruler and Compass ....
.






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what are the polygons? how many sides do each of them have?








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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a dodecagon is any polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 with twelve
12 (number)

12 is the natural number following 11 and preceding 13 .The word "twelve" is a native English language word that presumably arises from the Germanic languages compound twa-lif "two-leave", meaning that two is...
 sides and twelve angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s.

Regular dodecagon


It usually refers to a regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 dodecagon, having all sides of equal length and all angles equal to 150°. Its Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .

The area of a regular dodecagon with side a is given by:

Or, if R is the radius of a circumscribed circle,

And, if r is the radius of a inscribe
Inscribe

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Specifically, at all points where figures meet, their edges must lie tangent....
d circle,

Dodecagon construction

A regular dodecagon is constructible with compass and straightedge
Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization ruler and Compass ....
. The following is a 23-step animation illustrating one way it can be done. Notice that the compass radius is unaltered during steps 8 through 11.
Dodecagonconstructionani

Tilings


Here are 3 example periodic plane tilings that use dodecagons:

Tile 3bb

Semiregular tiling 3.12.12
Truncated hexagonal tiling

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....
Tile 46b

Semiregular tiling: 4.6.12
Great rhombitrihexagonal tiling

In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex ....
Dem3343tbc

A demiregular tiling:
3.3.4.12 & 3.3.3.3.3.3


Examples in use

  • When spelled uppercase, the outlines of the letters E
    E

    E is the fifth letter in the Latin alphabet. Its name in English language is spelled e , plural ees . The letter E is the most commonly used letter in the Czech language, Danish language, Dutch language, English language, French language, German language, Hungarian language, Latin language, Norwegian language, Spanish language...
     and H
    H

    H is the eighth letter in the Latin alphabet. Its name in both British English and American English is aitch , though it is also pronounced haitch in some dialects ....
     (and I
    I

    I is the ninth Letter of the Latin alphabet. Its English language name is i ....
     in a slab serif
    Slab serif

    In typography, a slab serif typeface is a type of serif typeface characterized by thick, block-like serifs. Serif terminals may be either blunt and angular , or rounded ....
     font) are all dodecagons.
  • The British threepenny bit was a regular dodecagon from 1937 to 1971, at which time it ceased to be legal tender. The Australian 50-cent coin, Fijian 50-cent, Tongan 50-seniti, and Solomon Island 50-cent coin are regular dodecagons. Croatian 25 kuna coin
    Croatian kuna

    The kuna is the currency of Croatia . The word "kuna" means "marten" in Croatian language since it is based on the use of marten pelts as units of value in medieval trading....
     has a dodecagonal shape as well. Until July 2005, a Romanian coin (5000 ROL
    Romanian leu

    The leu is the currency of Romania. It is subdivided into 100 bani . On 1 July 2005, Romania underwent a currency reform, switching from the previous leu to a new leu ....
    ) was also dodecagonal. The Canadian penny
    Penny (Canadian coin)

    In Canada, a penny is a coin worth one cent or of a Canadian dollar. According to the Royal Canadian Mint, the official national term of the coin is the "one-cent piece", but in practice the term penny or cent is universal....
     was dodecagonal from 1982 to 1996 as well as the South Vietnamese 20 Ðong coin until 1975. Zambian 50 ngwee (until 1992) and Malawian 50 tambala (until 1995) coins were dodecagonal. The Mexican 20 cent coin is also 12-sided.


See also

  • dodecagonal number
    Dodecagonal number

    A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for n is given by the formula 5n2 - 4n, with n > 0....
  • Dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
     - a regular polyhedron
    Polyhedron

    |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
     with 12 pentagon
    Pentagon

    In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
    al faces.
  • The E6 polytope can be drawn with an orthogonal projection inside a regular dodecagon.


External links

  • by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
  • With interactive animation