Hexagon - wikipedia Hexagon Not to be confused with Hexalgon. For other uses, see Hexagon ( disambiguation ). Regular hexagon A regular hexagon Type Regular polygon Edges and vertices 6 Schläfli symbol ( 6 ), t ( 3 ) Coxeter diagram Symmetry group Dihedral ( D ), order 2 × 6 Internal angle ( degrees ) 120 ° Dual polygon Self Properties Convex, cyclic, equilateral, isogonal, isotoxal In geometry, a hexagon ( from Greek ἕξ hex, `` six '' and γωνία, gonía, `` corner, angle '' ) is a six - sided polygon or 6 - gon. The total of the internal angles of any simple ( non-self - intersecting ) hexagon is 720 °. Contents 1 Regular hexagon 1.1 Parameters 2 Symmetry 2.1 A2 and G2 groups 3 Dissection 4 Related polygons and tilings 5 Hexagonal structures 6 Tesselations by hexagons 7 Hexagon inscribed in a conic section 7.1 Cyclic hexagon 8 Hexagon tangential to a conic section 9 Equilateral triangles on the sides of an arbitrary hexagon 10 Skew hexagon 10.1 Petrie polygons 11 Convex equilateral hexagon 11.1 Polyhedra with hexagons 12 Hexagons : natural and human - made 13 See also 14 References 15 External links Regular hexagon A regular hexagon has Schläfli symbol ( 6 ) and can also be constructed as a truncated equilateral triangle, t ( 3 ), which alternates two types of edges. A step - by - step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid 's Elements, Book IV, Proposition 15 : this is possible as 6 = ( displaystyle = ) 2 × 3, a product of a power of two and distinct Fermat primes. When the side length AB is given, then you draw around the point A and around the point B a circular arc. The intersection M is the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic ( has a circumscribed circle ) and tangential ( has an inscribed circle ). The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 ( displaystyle ( \tfrac ( 2 ) ( sqrt ( 3 ) ) ) ) times the apothem ( radius of the inscribed circle ). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries ( rotational symmetry of order six ) and 6 reflection symmetries ( six lines of symmetry ), making up the dihedral group D. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane ( three hexagons meeting at every vertex ), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral. Parameters The maximal diameter ( which corresponds to the long diagonal of the hexagon ), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle ( separation of parallel sides, flat - to - flat distance, short diagonal or height when resting on a flat base ), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor : 1 2 d = r = cos ⁡ ( 30 ∘ ) R = 3 2 R = 3 2 t ( displaystyle ( \frac ( 1 ) ( 2 ) ) d = r = cos ( 30 ^ ( circ ) ) R = ( \frac ( sqrt ( 3 ) ) ( 2 ) ) R = ( \frac ( sqrt ( 3 ) ) ( 2 ) ) t ) and, similarly, d = 3 2 D. ( displaystyle d = ( \frac ( sqrt ( 3 ) ) ( 2 ) ) D. ) The area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. ( displaystyle ( \begin ( aligned ) A& = ( \frac ( 3 ( sqrt ( 3 ) ) ) ( 2 ) ) R ^ ( 2 ) = 3Rr = 2 ( sqrt ( 3 ) ) r ^ ( 2 ) & = ( \frac ( 3 ( sqrt ( 3 ) ) ) ( 8 ) ) D ^ ( 2 ) = ( \frac ( 3 ) ( 4 ) ) Dd = ( \frac ( sqrt ( 3 ) ) ( 2 ) ) d ^ ( 2 ) & approx 2.598 R ^ ( 2 ) approx 3.464 r ^ ( 2 ) & approx 0.6495 D ^ ( 2 ) approx 0.866 d ^ ( 2 ). end ( aligned ) ) ) For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p = 6 R = 4 r 3 ( displaystyle ( ) = 6R = 4r ( sqrt ( 3 ) ) ), so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. ( displaystyle ( \begin ( aligned ) A& = ( \frac ( ap ) ( 2 ) ) & = ( \frac ( r cdot 4r ( sqrt ( 3 ) ) ) ( 2 ) ) = 2r ^ ( 2 ) ( sqrt ( 3 ) ) & approx 3.464 r ^ ( 2 ). end ( aligned ) ) ) The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 ( displaystyle ( \tfrac ( 3 ( sqrt ( 3 ) ) ) ( 2 pi ) ) approx 0.8270 ) of its circumscribed circle. If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C, then PE + PF = PA + PB + PC + PD. Symmetry The six lines of reflection of a regular hexagon, with Dih or r12 symmetry, order 12. The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal ) or edges ( p for perpendiculars ) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r12 and no symmetry is labeled a1. The regular hexagon has Dih symmetry, order 12. There are 3 dihedral subgroups : Dih, Dih, and Dih, and 4 cyclic subgroups : Z, Z, Z, and Z. These symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. r12 is full symmetry, and a1 is no symmetry. d6, a isogonal hexagon constructed by four mirrors can alternate long and short edges, and p6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can seen as directed edges. Example hexagons by symmetry r12 regular i4 d6 isotoxal g6 directed p6 isogonal d2 g2 general parallelogon p2 g3 a1 Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations. p6m ( * 632 ) cmm ( 2 * 22 ) p2 ( 2222 ) p31m ( 3 * 3 ) pmg ( 22 * ) pg ( × × ) r12 i4 g2 d2 d2 p2 a1 A2 and g2 groups A2 group roots G2 group roots The 6 roots of the simple Lie group A2, represented by a Dynkin diagram, are in a regular hexagonal pattern. The two simple roots have a 120 ° angle between them. The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150 ° angle between them. Dissection 6 - cube projection 10 rhomb dissection 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. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids. Dissection of hexagons into 3 rhombs and parallelograms 2D Rhombs Parallelograms Regular ( 6 ) Hexagonal parallelogons 3D Square faces Rectangular faces Cube Rectangular cuboid Related polygons and tilings A regular hexagon has Schläfli symbol ( 6 ). A regular hexagon is a part the regular hexagonal tiling, ( 6, 3 ), with 3 hexagonal around each vertex. A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t ( 3 ). Seen with two types ( colors ) of edges, this form only has D symmetry. A truncated hexagon, t ( 6 ), is a dodecagon, ( 12 ), alternating 2 types ( colors ) of edges. An alternated hexagon, h ( 6 ), is a equilateral triangle, ( 3 ). A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into 6 equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling. A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling. Regular ( 6 ) Truncated t ( 3 ) = ( 6 ) Hypertruncated triangles Stellated Star figure 2 ( 3 ) Truncated t ( 6 ) = ( 12 ) Alternated h ( 6 ) = ( 3 ) A concave hexagon A self - intersecting hexagon ( star polygon ) Dissected ( 6 ) Extended Central ( 6 ) in ( 12 ) A skew hexagon, within cube Hexagonal structures Giant 's Causeway closeup From bees ' honeycombs to the Giant 's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest number of hexagons. This means that honeycombs require less wax to construct and gain lots of strength under compression. Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3 - space by translation. Hexagonal prism tessellations Form Hexagonal tiling Hexagonal prismatic honeycomb Regular Parallelogonal Tesselations by hexagons Main article : Hexagonal tiling In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. Hexagon inscribed in a conic section Pascal 's theorem ( also known as the `` Hexagrammum Mysticum Theorem '' ) states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the `` Pascal line '' of that configuration. Cyclic hexagon The Lemoine hexagon is a cyclic hexagon ( one inscribed in a circle ) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf. If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent. If a hexagon has vertices on the circumcircle of an acute triangle at the six points ( including three triangle vertices ) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle. Hexagon tangential to a conic section Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon 's theorem states that the three main diagonals AD, BE, and CF intersect at a single point. In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f, a + c + e = b + d + f. ( displaystyle a + c + e = b + d + f. ) Equilateral triangles on the sides of an arbitrary hexagon Equilateral triangles on the sides of an arbitrary hexagon If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle. Skew hexagon A regular skew hexagon seen as edges ( black ) of a triangular antiprism, symmetry D, ( 2, 6 ), ( 2 * 3 ), order 12. A skew hexagon is a skew polygon with 6 vertices and edges but not existing on the same plane. The interior of such an hexagon is not generally defined. A skew zig - zag hexagon has vertices alternating between two parallel planes. A regular skew hexagon is vertex - transitive with equal edge lengths. In 3 - dimensions it will be a zig - zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D, ( 2, 6 ) symmetry, order 12. The cube and octahedron ( same as triangular antiprism ) have regular skew hexagons as petrie polygons. Skew hexagons on 3-fold axes Cube Octahedron Petrie polygons The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections : 4D 5D 3 - 3 duoprism 3 - 3 duopyramid 5 - simplex Convex equilateral hexagon A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon ( one with all sides equal ) with common side a, there exists a principal diagonal d such that d 1 a ≤ 2 ( displaystyle ( \frac ( d_ ( 1 ) ) ( a ) ) leq 2 ) and a principal diagonal d such that d 2 a > 3. ( displaystyle ( \frac ( d_ ( 2 ) ) ( a ) ) > ( sqrt ( 3 ) ). ) Polyhedra with hexagons There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to `` fold up ''. The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron ( of soccer ball and fullerene fame ), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and. Hexagons in Archimedean solids Tetrahedral Octahedral Icosahedral truncated tetrahedron truncated octahedron truncated cuboctahedron truncated icosahedron truncated icosidodecahedron There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G ( 2, 0 ) : Hexagons in Goldberg polyhedra Tetrahedral Octahedral Icosahedral Chamfered tetrahedron Chamfered cube Chamfered dodecahedron There are also 9 Johnson solids with regular hexagons : Johnson solids and near - misses with hexagons triangular cupola elongated triangular cupola gyroelongated triangular cupola augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism augmented truncated tetrahedron triangular hebesphenorotunda Truncated triakis tetrahedron Prismoids with hexagons Hexagonal prism Hexagonal antiprism Hexagonal pyramid Tilings with regular hexagons Regular 1 - uniform ( 6, 3 ) r ( 6, 3 ) rr ( 6, 3 ) tr ( 6, 3 ) 2 - uniform tilings Hexagons : natural and human - made The ideal crystalline structure of graphene is a hexagonal grid. Assembled E-ELT mirror segments A beehive honeycomb The scutes of a turtle 's carapace North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini ( 1 ) ( 2 ) ( 3 ) Micrograph of a snowflake Benzene, the simplest aromatic compound with hexagonal shape. Hexagonal order of bubbles in a foam. Crystal structure of a molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834 - 1839. Naturally formed basalt columns from Giant 's Causeway in Northern Ireland ; large masses must cool slowly to form a polygonal fracture pattern An aerial view of Fort Jefferson in Dry Tortugas National Park The James Webb Space Telescope mirror is composed of 18 hexagonal segments. Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France aka the `` métropole '' as opposed to the overseas territories such as Guadeloupe, Martinique or French Guiana. Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals Hexagonal barn The Hexagon, a hexagonal theatre in Reading, Berkshire Władysław Gliński 's hexagonal chess Pavilion in the Taiwan Botanical Gardens Hexagonal window See also 24 - cell : a four - dimensional figure which, like the hexagon, has orthoplex facets, is self - dual and tessellates Euclidean space Hexagonal crystal system Hexagonal number Hexagonal tiling : a regular tiling of hexagons in a plane Hexagram : 6 - sided star within a regular hexagon Unicursal hexagram : single path, 6 - sided star, within a hexagon Honeycomb conjecture References Jump up ^ Wenninger, Magnus J. ( 1974 ), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595. 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 ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p. 141 Jump up ^ Cartensen, Jens, `` About hexagons '', Mathematical Spectrum 33 ( 2 ) ( 2000 -- 2001 ), 37 -- 40. Jump up ^ Nikolaos Dergiades, `` Dao 's theorem on six circumcenters associated with a cyclic hexagon '', Forum Geometricorum 14, 2014, 243 -- 246. http://forumgeom.fau.edu/FG2014volume14/FG201424index.html Jump up ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 ( orig. 1960 ). Jump up ^ Gutierrez, Antonio, `` Hexagon, Inscribed Circle, Tangent, Semiperimeter '', ( 4 ), Accessed 2012 - 04 - 17. Jump up ^ Dao Thanh Oai ( 2015 ), `` Equilateral triangles and Kiepert perspectors in complex numbers '', Forum Geometricorum 15, 105 -- 114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html Jump up ^ Inequalities proposed in `` Crux Mathematicorum '', ( 5 ). External links Look up hexagon in Wiktionary, the free dictionary. Weisstein, Eric W. `` Hexagon ''. MathWorld. This article 's use of external links may not follow Wikipedia 's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. ( November 2017 ) Definition and properties of a hexagon with interactive animation and construction with compass and straightedge. Cymatics -- Hexagonal shapes occurring within water sound images Cassini Images Bizarre Hexagon on Saturn Saturn 's Strange Hexagon A hexagonal feature around Saturn 's North Pole `` Bizarre Hexagon Spotted on Saturn '' -- from Space.com ( 27 March 2007 ) 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 Fundamental convex regular and uniform polytopes in dimensions 2 -- 10 Family I ( p ) / D E / E / E / F / G Regular polygon Triangle Square p - gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform 4 - polytope 5 - cell 16 - cell Tesseract Demitesseract 24 - cell 120 - cell 600 - cell Uniform 5 - polytope 5 - simplex 5 - orthoplex 5 - cube 5 - demicube Uniform 6 - polytope 6 - simplex 6 - orthoplex 6 - cube 6 - demicube 1 2 Uniform 7 - polytope 7 - simplex 7 - orthoplex 7 - cube 7 - demicube 1 2 3 Uniform 8 - polytope 8 - simplex 8 - orthoplex 8 - cube 8 - demicube 1 2 4 Uniform 9 - polytope 9 - simplex 9 - orthoplex 9 - cube 9 - demicube Uniform 10 - polytope 10 - simplex 10 - orthoplex 10 - cube 10 - demicube Uniform n - polytope n - simplex n - orthoplex n - cube n - demicube 1 2 k n - pentagonal polytope Topics : Polytope families Regular polytope List of regular polytopes and compounds Retrieved from `` https://en.wikipedia.org/w/index.php?title=Hexagon&oldid=836225795 '' Categories : Polygons 6 ( number ) Constructible polygons Elementary shapes Hidden categories : Wikipedia external links cleanup from November 2017 Wikipedia spam cleanup from November 2017 All articles with dead external links Articles with dead external links from May 2010 Talk Contents About Wikipedia Afrikaans অসমীয়া Asturianu Башҡортса Български Català Čeština ChiShona Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Gaelg Galego 客家 語 / Hak - kâ - ngî 한국어 Հայերեն हिन्दी Ido Bahasa Indonesia Italiano עברית Basa Jawa Kreyòl ayisyen Latina Latviešu Lëtzebuergesch Lietuvių Lumbaart Magyar Македонски മലയാളം Bahasa Melayu Монгол 日本 語 Nordfriisk Norsk Norsk nynorsk Nouormand پښتو Polski Português Română Русский Scots Sicilianu Simple English Slovenčina Slovenščina کوردی Српски / srpski Srpskohrvatski / српскохрватски Basa Sunda Suomi Svenska Tagalog தமிழ் తెలుగు ไทย Türkçe Українська Tiếng Việt Winaray 粵語 中文 63 more Edit links This page was last edited on 13 April 2018, at 13 : 07. 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Question: how many lines of symmetry does a hexagon has
Answer: 9