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Regular dodecagon
Edges and vertices 12
Schläfli symbols {12}
t{6}
Coxeter–Dynkin diagrams Image:CDW_ring.pngImage:CDW_12.pngImage:CDW_dot.png
Image:CDW_ring.pngImage:CDW_6.pngImage:CDW_ring.png
Symmetry group Dihedral (D12)
Area
(with t=edge length)		 A = 3 \cot\left( \frac{\pi}{12} \right) t^2 = 3 \left( 2+\sqrt{3} \right) t^2 \simeq 11.19615242 t^2.
Internal angle
(degrees)		 150°

In geometry, a dodecagon is any polygon with twelve sides and twelve angles. For edge in Graph theory, see Edge (graph theory In Geometry, an edge is a one-dimensional Line segment joining In Geometry, a vertex (plural "vertices" is a special kind of point. In Mathematics, the Schläfli symbol is a notation of the form {pqr The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle This article describes the unit of angle For other meanings see Degree. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called


Contents

Regular dodecagon

It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°. General properties These properties apply to both convex and star regular polygons Its Schläfli symbol is {12}. In Mathematics, the Schläfli symbol is a notation of the form {pqr

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

A = 3 \cot\left( \frac{\pi}{12} \right) a^2 = 3 \left( 2+\sqrt{3} \right) a^2 \simeq 11.19615242 a^2.

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

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

And, if r is the radius of a inscribed circle,

A = 12 \tan\left( \frac{\pi}{12}\right) r^2 = 12 \left( 2-\sqrt{3} \right) r^2 \simeq 3.2153903 r^2.

Dodecagon construction

A regular dodecagon is constructible with compass and straightedge. In Geometry, an inscribed Planar Shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles 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.

Dodecagon Construction Animation

Tilings

Here are 3 example periodic plane tilings that use dodecagons:

Tile 3bb.svg
Semiregular tiling 3.12.12
Semiregular tiling: 4.6.12
A demiregular tiling:
3. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps In Geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. In Geometry, the Great rhombitrihexagonal tiling (or Omnitruncated trihexagonal tiling) is a semiregular tiling of the Euclidean plane Plane tilings by Regular polygons have been widely used since antiquity 3. 4. 12 & 3. 3. 3. 3. 3. 3

Examples in use

See also

External links

Dictionary

dodecagon

-noun

  1. (geometry) a polygon with twelve edges and twelve angles
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