Dodecagon - wikipedia Dodecagon Regular dodecagon A regular dodecagon Type Regular polygon Edges and vertices 12 Schläfli symbol ( 12 ), t ( 6 ), tt ( 3 ) Coxeter diagram Symmetry group Dihedral ( D ), order 2 × 12 Internal angle ( degrees ) 150 ° Dual polygon Self Properties Convex, cyclic, equilateral, isogonal, isotoxal In geometry, a dodecagon or 12 - gon is any twelve - sided polygon. Contents 1 Regular dodecagon 1.1 Area 1.2 Perimeter 2 Dodecagon construction 3 Dissection 4 Symmetry 5 Occurrence 5.1 Tiling 6 Skew dodecagon 6.1 Petrie polygons 7 Related figures 8 Examples in use 9 See also 10 Notes 11 External links Regular dodecagon A regular 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 ( 12 ) and can be constructed as a truncated hexagon, t ( 6 ), or a twice - truncated triangle, tt ( 3 ). The internal angle at each vertex of a regular dodecagon is 150 °. Area The area of a regular dodecagon of side length a is given by : A = 3 cot ⁡ ( π 12 ) a 2 = 3 ( 2 + 3 ) a 2 ≃ 11.19615242 a 2 ( displaystyle ( \begin ( aligned ) A& = 3 cot left ( ( \frac ( pi ) ( 12 ) ) \right ) a ^ ( 2 ) = 3 left ( 2 + ( sqrt ( 3 ) ) \right ) a ^ ( 2 ) & simeq 11.19615242 , a ^ ( 2 ) end ( aligned ) ) ) And in terms of the apothem r ( see also inscribed figure ), the area is : A = 12 tan ⁡ ( π 12 ) r 2 = 12 ( 2 − 3 ) r 2 ≃ 3.2153903 r 2 ( displaystyle ( \begin ( aligned ) A& = 12 \tan left ( ( \frac ( pi ) ( 12 ) ) \right ) r ^ ( 2 ) = 12 left ( 2 - ( sqrt ( 3 ) ) \right ) r ^ ( 2 ) & simeq 3.2153903 , r ^ ( 2 ) end ( aligned ) ) ) In terms of the circumradius R, the area is : A = 6 sin ⁡ ( π 6 ) R 2 = 3 R 2 ( displaystyle A = 6 sin left ( ( \frac ( pi ) ( 6 ) ) \right ) R ^ ( 2 ) = 3R ^ ( 2 ) ) The span 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 = 3 a S ( displaystyle A = 3aS ) This can be verified with the trigonometric relationship : S = a ( 1 + 2 cos ⁡ 30 ∘ + 2 cos ⁡ 60 ∘ ) ( displaystyle S = a ( 1 + 2 cos ( 30 ^ ( circ ) ) + 2 cos ( 60 ^ ( circ ) ) ) ) Perimeter The perimeter of a regular dodecagon in terms of circumradius is : p = 24 R tan ⁡ ( π 12 ) = 12 R 2 − 3 ≃ 6.21165708246 R ( displaystyle ( \begin ( aligned ) p& = 24R \tan left ( ( \frac ( pi ) ( 12 ) ) \right ) = 12R ( sqrt ( 2 - ( sqrt ( 3 ) ) ) ) & simeq 6.21165708246 , R end ( aligned ) ) ) The perimeter in terms of apothem is : p = 24 r tan ⁡ ( π 12 ) = 24 r ( 2 − 3 ) ≃ 6.43078061835 r ( displaystyle ( \begin ( aligned ) p& = 24r \tan left ( ( \frac ( pi ) ( 12 ) ) \right ) = 24r ( 2 - ( sqrt ( 3 ) ) ) & simeq 6.43078061835 , r end ( aligned ) ) ) This coefficient is double the coefficient found in the apothem equation for area. Dodecagon construction As 12 = 2 × 3, regular dodecagon is constructible using compass - and - straightedge construction : Construction of a regular dodecagon at a given circumcircle Construction of a regular dodecagon at a given side length, animation. ( The construction is very similar to that of octagon at a given side length. ) Dissection 12 - cube 60 rhomb dissection Isotoxal dodecagon Coxeter states that every zonogon ( a 2m - 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 projection of a 6 - cube, with 15 of 240 faces. The sequence OEIS sequence A006245 defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection. Dissection into 15 rhombs 6 - cube 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. Other dissections Regular pattern blocks Symmetry The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives d ( diagonal ) with mirror lines through vertices, p with mirror lines through edges ( perpendicular ), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons. The regular dodecagon has Dih 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. show Example dodecagons by symmetry r24 d12 g12 p12 i8 d6 g6 p6 d4 g4 p4 g3 d2 g2 p2 a1 Occurrence Tiling A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways : 3.12. 12 4.6. 12 3.3. 4.12 3.4. 3.12 Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration : 1 - uniform 2 - uniform 3.12. 12 4.6. 12 3.12. 12 ; 3.4. 3.12 Skew dodecagon A regular skew dodecagon seen as zig - zagging edges of a hexagonal antiprism. 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 D, ( 2, 10 ) symmetry, order 20. The dodecagrammic antiprism, s ( 2, 24 / 5 ) and dodecagrammic crossed - antiprism, s ( 2, 24 / 7 ) also have regular skew dodecagons. Petrie polygons The regular dodecagon is the Petrie polygon for many higher - dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24 - cell, snub 24 - cell, 6 - 6 duoprism, 6 - 6 duopyramid. In 6 dimensions 6 - cube, 6 - orthoplex, 2, 1. It is also the Petrie polygon for the grand 120 - cell and great stellated 120 - cell. show Regular skew dodecagons in higher dimensions 2G ( 4D ) 24 - cell Snub 24 - cell 6 - 6 duopyramid 6 - 6 duoprism 11 - simplex ( 4 ) 6 - orthoplex 6 - cube Related figures A dodecagram is a 12 - sided star polygon, represented by symbol ( 12 / n ). There is one regular star polygon : ( 12 / 5 ), using the same vertices, but connecting every fifth point. There are also three compounds : ( 12 / 2 ) is reduced to 2 ( 6 ) as two hexagons, and ( 12 / 3 ) is reduced to 3 ( 4 ) as three squares, ( 12 / 4 ) is reduced to 4 ( 3 ) as four triangles, and ( 12 / 6 ) is reduced to 6 ( 2 ) as six degenerate digons. show Stars and compounds n 5 6 Form Polygon Compounds Star polygon Compound Image ( 12 / 1 ) = ( 12 ) ( 12 / 2 ) or 2 ( 6 ) ( 12 / 3 ) or 3 ( 4 ) ( 12 / 4 ) or 4 ( 3 ) ( 12 / 5 ) ( 12 / 6 ) or 6 ( 2 ) 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 ( 6 ) = ( 12 ). A quasitruncated hexagon, inverted as ( 6 / 5 ), is a dodecagram : t ( 6 / 5 ) = ( 12 / 5 ). show Vertex - transitive truncations of the hexagon Quasiregular Isogonal Quasiregular t ( 6 ) = ( 12 ) t ( 6 / 5 ) = ( 12 / 5 ) 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 Vera Cruz church in Segovia The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military 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 ''. A 1942 British threepence, reverse Regular dodecagonal coins include : British threepenny bit from 1937 to 1971, when it ceased to be legal tender. British One Pound Coin, introduced in 2017. 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 20 đồng, 1968 -- 1975 Zambian 50 ngwee, 1969 -- 1992 Malawian 50 tambala, 1986 -- 1995 Mexican 20 centavos, 1992 - 2009 In the Philippines, in local carnivals ( peryahan ), ferris wheels usually with 12 seats or gondolas See also Dodecagonal number Dodecahedron -- a regular polyhedron with 12 pentagonal faces. Dodecagram Notes Jump up ^ See also Kürschák 's geometric proof on the Wolfram Demonstration Project Jump up ^ Plane Geometry : Experiment, Classification, Discovery, Application by Clarence Addison Willis B., ( 1922 ) Blakiston 's Son & Company, p. 249 ( 1 ) Jump up ^ Elements of geometry by John Playfair, William Wallace, John Davidsons, ( 1814 ) Bell & Bradfute, p. 243 ( 2 ) Jump up ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p. 141 Jump up ^ `` Doin ' Da ' Dodeca ' '' on mathforum.org Jump up ^ John H. Conway, Heidi Burgiel, Chaim Goodman - Strauss, ( 2008 ) The Symmetries of Things, ISBN 978 - 1 - 56881 - 220 - 5 ( Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275 - 278 ) Jump up ^ The Lighter Side of Mathematics : Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, ( 1994 ), Metamorphoses of polygons, Branko Grünbaum External links Weisstein, Eric W. `` Dodecagon ''. MathWorld. Kürschak 's Tile and Theorem Definition and properties of a dodecagon With interactive animation Polygons Regular List 1 -- 10 sides Monogon Digon Triangle Equilateral Isosceles Quadrilateral Square Rectangle Rhombus Parallelogram Trapezoid Kite Pentagon Hexagon Heptagon Octagon Nonagon ( Enneagon ) Decagon 11 -- 20 sides Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon 21 -- 100 sides ( selected ) Icosidigon ( 22 ) Icositetragon ( 24 ) Icosihexagon ( 26 ) Icosioctagon ( 28 ) Triacontagon ( 30 ) Triacontadigon ( 32 ) Triacontatetragon ( 34 ) Tetracontagon ( 40 ) Tetracontadigon ( 42 ) Tetracontaoctagon ( 48 ) Pentacontagon ( 50 ) Hexacontagon ( 60 ) Hexacontatetragon ( 64 ) Heptacontagon ( 70 ) Octacontagon ( 80 ) Enneacontagon ( 90 ) Enneacontahexagon ( 96 ) Hectogon ( 100 ) > 100 sides 120 - gon 257 - gon 360 - gon Chiliagon ( 1,000 ) Myriagon ( 10,000 ) 65537 - gon Megagon ( 1,000,000 ) Apeirogon ( ∞ ) Star polygons ( 5 -- 12 sides ) Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram Retrieved from `` https://en.wikipedia.org/w/index.php?title=Dodecagon&oldid=849067946 '' Categories : Polygons Constructible polygons Talk Contents About Wikipedia Asturianu Български Català Čeština Deutsch Ελληνικά Español Esperanto Euskara فارسی Français Galego 客家 語 / Hak - kâ - ngî 한국어 Հայերեն Bahasa Indonesia Italiano ქართული Kreyòl ayisyen Magyar Македонски Bahasa Melayu Nederlands 日本 語 Norsk Norsk nynorsk Português Română Русский Simple English Slovenščina Српски / srpski Svenska ไทย Türkçe Українська اردو 粵語 中文 31 more Edit links This page was last edited on 6 July 2018, at 08 : 43 ( UTC ). 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Question: what is the name for a 12 sided polygon
Answer: dodecagon