dodecagon

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In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a dodecagon or 12-gon is any twelve-sided
polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region (mathematic ...

.

# Regular dodecagon

A
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ...
dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who ...
and can be constructed as a truncated
hexagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

, t, or a twice-truncated
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The b ...

, tt. The internal angle at each vertex of a regular dodecagon is 150°.

## Area

The
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the am ...

of a regular dodecagon of side length ''a'' is given by: :$\begin A & = 3 \cot\left\left(\frac \right\right) a^2 = 3 \left\left(2+\sqrt \right\right) a^2 \\ & \simeq 11.19615242\,a^2 \end$ And in terms of the
apothem Apothem of a hexagon The apothem (sometimes abbreviated as apo) of a regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, equiangular (all angles are equal in measure) and Equilateral polygon, equi ...

inscribed figure{{unreferenced, date=August 2012 frame, Inscribed circles of various polygons An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid is one that is enclosed by and "fits snugly" insid ...
), the area is: :$\begin A & = 12 \tan\left\left(\frac\right\right) r^2 = 12 \left\left(2-\sqrt \right\right) r^2 \\ & \simeq 3.2153903\,r^2 \end$ In terms of the
circumradius In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
''R'', the area is: :$A = 6 \sin\left\left(\frac\right\right) R^2 = 3 R^2$ The span ''S'' of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is: :$A = 3aS$ This can be verified with the trigonometric relationship: :$S = a\left(1+ 2\cos + 2\cos\right)$

## Perimeter

The
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length ( one-dimensional). The perimeter of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathema ...

of a regular dodecagon in terms of circumradius is: :$\begin p & = 24R \tan\left\left(\frac\right\right) = 12R \sqrt\\ & \simeq 6.21165708246\,R \end$ The perimeter in terms of apothem is: :$\begin p & = 24r \tan\left\left(\frac\right\right) = 24r\left(2-\sqrt\right)\\ & \simeq 6.43078061835\,r \end$ This coefficient is double the coefficient found in the apothem equation for area.

# Dodecagon construction

As 12 = 22 × 3, regular dodecagon is constructible using
compass-and-straightedge construction Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ...
:

# Dissection

Coxeter Harold Scott MacDonald "Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington ...
states that every
zonohedronA zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a zonogon, polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski addition, Minkowski sum of a set of ...
(a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular dodecagon'', ''m''=6, and it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs. This decomposition is based on a
Petrie polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
projection of a 6-cube, with 15 of 240 faces. The sequence OEIS sequence defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons. They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.

# Symmetry

The ''regular dodecagon'' has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges.

# Occurrence

## Tiling

A regular dodecagon can Euclidean tilings of convex regular polygons, fill a plane vertex with other regular polygons in 4 ways: Here are 3 example Tiling by regular polygons, periodic plane tilings that use regular dodecagons, defined by their vertex configuration:

# Skew dodecagon

A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such an dodecagon is not generally defined. A ''skew zig-zag dodecagon'' has vertices alternating between two parallel planes. A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, [2+,10] symmetry, order 20. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons.

## Petrie polygons

The regular dodecagon is the
Petrie polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. In 6 dimensions 6-cube, 6-orthoplex, 2 21 polytope, 221, 1 22 polytope, 122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.

# Related figures

A dodecagram is a 12-sided star polygon, represented by symbol . There is one regular star polygon: , using the same vertices, but connecting every fifth point. There are also three compounds: is reduced to 2 as two
hexagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t=. A quasitruncated hexagon, inverted as , is a dodecagram: t=.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', Branko Grünbaum

# Examples in use

In block capitals, the letters E, H and X (and I in a slab serif font) have dodecagonal outlines. A cross is a dodecagon, as is the logo for the Chevrolet automobile division. The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military Watchtower (fortification), 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". Regular dodecagonal coins include: *Threepence (British coin), British threepenny bit from 1937 to 1971, when it ceased to be legal tender. *One pound (British coin), British One Pound Coin, introduced in 2017. *Australian 50-cent coin *Coins of the Fijian dollar, Fijian 50 cents *Tongan paʻanga, Tongan 50-seniti, since 1974 *Solomon Islands dollar, Solomon Islands 50 cents *Croatian kuna, Croatian 25 kuna *Romanian leu, Romanian 5000 lei, 2001–2005 *Penny (Canadian coin), Canadian penny, 1982–1996 *South Vietnamese đồng, South Vietnamese 20 đồng, 1968–1975 *Zambian kwacha, Zambian 50 ngwee, 1969–1992 *Malawian kwacha, Malawian 50 tambala, 1986–1995 *Mexican peso, Mexican 20 centavos, 1992-2009

*Dodecagonal number *Dodecahedron – any polyhedron with 12 faces. *Dodecagram