Regular hexagon  

Type  Regular polygon 
Edges and vertices  6 
Schläfli symbol  {6}, t{3} 
Coxeter–Dynkin diagrams  
Symmetry group  Dihedral (D_{6}), order 2×6 
Internal angle (degrees)  120° 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
Dual polygon  Self 
In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a sixsided polygon.^{ [1] } The total of the internal angles of any simple (nonselfintersecting) hexagon is 720°.
A regular hexagon has Schläfli symbol {6}^{ [2] } and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.
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 or circumcircle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D_{6}. 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.
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, flattoflat 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:
The area of a regular hexagon
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, so
The regular hexagon fills the fraction 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 circumcircle between B and C, then PE + PF = PA + PB + PC + PD.
It follows from the ratio of circumradius to inradius that the heighttowidth ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.
For an arbitrary point in the plane of a regular hexagon with circumradius , whose distances to the centroid of the regular hexagon and its six vertices are and respectively, we have^{ [3] }
If are the distances from the vertices of a regular hexagon to any point on its circumcircle, then ^{ [3] }
Example hexagons by symmetry  


The regular hexagon has D_{6} symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D_{6}), 2 dihedral: (D_{3,} D_{2}), 4 cyclic: (Z_{6}, Z_{3}, Z_{2}, Z_{1}) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.^{ [4] }r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, 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 be seen as directed edges.
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 
Dih_{6}  Dih_{2}  Z_{2}  Dih_{1}  Z_{1} 
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.
6cube projection  12 rhomb dissection  

Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.^{ [5] } 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 three rhombs and parallelograms  

2D  Rhombs  Parallelograms  
Regular {6}  Hexagonal parallelogons  
3D  Square faces  Rectangular faces  
Cube  Rectangular cuboid 
A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces 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_{3} symmetry.
A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an 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 six 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} 

Crossed hexagon  A concave hexagon  A selfintersecting hexagon (star polygon)  Extended Central {6} in {12}  A skew hexagon, within cube  Dissected {6}  projection octahedron  Complete graph 

There are six selfcrossing hexagons with the vertex arrangement of the regular hexagon:
Dih_{2}  Dih_{1}  Dih_{3}  

Figureeight  Centerflip  Unicursal  Fishtail  Doubletail  Tripletail 
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 hexagons. This means that honeycombs require less wax to construct and gain much 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 3space by translation.
Form  Hexagonal tiling  Hexagonal prismatic honeycomb 

Regular  
Parallelogonal 
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.
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.
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.^{ [6] }
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.^{ [7] }
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.^{ [8] }^{: p. 179 }
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,^{ [9] }
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.^{ [10] }^{: Thm. 1 }
A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zigzag hexagon has vertices alternating between two parallel planes.
A regular skew hexagon is vertextransitive with equal edge lengths. In three dimensions it will be a zigzag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D_{3d}, [2^{+},6] symmetry, order 12.
The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Cube  Octahedron 
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  

33 duoprism  33 duopyramid  5simplex 
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^{ [11] }^{: p.184, #286.3 } a principal diagonal d_{1} such that
and a principal diagonal d_{2} such that
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:
Prismoids with hexagons  

Hexagonal prism  Hexagonal antiprism  Hexagonal pyramid 
Tilings with regular hexagons  

Regular  1uniform  
{6,3}  r{6,3}  rr{6,3}  tr{6,3}  
2uniform tilings  
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertextransitive but also edgetransitive. It is radially equilateral.
In geometry, a cube is a threedimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
In geometry, an octagon is an eightsided polygon or 8gon.
In geometry, a decagon is a tensided polygon or 10gon. The total sum of the interior angles of a simple decagon is 1440°.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, an icosagon or 20gon is a twentysided polygon. The sum of any icosagon's interior angles is 3240 degrees.
In geometry, a dodecagon, or 12gon, is any twelvesided polygon.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equallength adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
In geometry, a triacontagon or 30gon is a thirtysided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
In geometry, a pentadecagon or pentakaidecagon or 15gon is a fifteensided polygon.
In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
In mathematics, a hexadecagon is a sixteensided polygon.
In geometry, an octadecagon or 18gon is an eighteensided polygon.
In geometry, a pentagon is any fivesided polygon or 5gon. The sum of the internal angles in a simple pentagon is 540°.
In geometry, an icositetragon or 24gon is a twentyfoursided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.
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