In geometry, a hexagon (from Ancient Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a sixsided polygon or 6gon. The total of the internal angles of any simple polygon (nonselfintersecting) hexagon is 720°.
A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic polygon (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 $\backslash tfrac\{2\}\{\backslash sqrt\{3\}\}$ times the apothem (radius of the inscribed figure). All internal 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 , regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing . 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 area of a regular hexagon
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{align}
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$\{\}\; =\; 6R\; =\; 4r\backslash sqrt\{3\}$, so
A &= \frac{ap}{2} \\ &= \frac{r \cdot 4r\sqrt{3}}{2} = 2r^2\sqrt{3} \\ &\approx 3.464 r^2.\end{align}
The regular hexagon fills the fraction $\backslash tfrac\{3\backslash sqrt\{3\}\}\{2\backslash pi\}\; \backslash approx\; 0.8270$ 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 circumcircle between B and C, then .
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.
If $d\_i$ are the distances from the vertices of a regular hexagon to any point on its circumscircle, then
{ class=wikitable 
The regular hexagon has Dih_{6} symmetry, order 12. There are three dihedral subgroups: Dih_{3}, Dih_{2}, and Dih_{1}, and four cyclic group subgroups: Z_{6}, Z_{3}, Z_{2}, and Z_{1}.
These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278) r12 is full symmetry, and a1 is no symmetry. p6, an isogonal figure hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal figure hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are dual polygon 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 .
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 .
Hexagons of symmetry g2, i4, and r12, as can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.
A2 group roots  G2 group roots 
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.
Regular {6}  Hexagonal  
Cube  Rectangular cuboid 
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 stellation 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 and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.
+ Selfintersecting hexagons with regular vertices !colspan=3  Dih_{2} !colspan=2  Dih_{1} !Dih_{3}  
Figureeight  Centerflip  Unicursal  Fishtail  Doubletail  Tripletail 
Irregular hexagons with parallel opposite edges are called and can also tile the plane by translation. In three dimensions, with parallel opposite faces are called and these can tessellate 3space by translation.
+ Hexagonal prism tessellations !Form !Hexagonal tiling !Hexagonal prismatic honeycomb 
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 .Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
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 lines.
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.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [1] , Accessed 20120417.
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.
+Skew hexagons on 3fold axes  
Cube  Octahedron 
33 duoprism  33 duopyramid  5simplex 
and a principal diagonal d_{2} such that
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):
Chamfered tetrahedron  Chamfered cube  Chamfered dodecahedron 
There are also 9 with regular 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 
Hexagonal prism  Hexagonal antiprism  Hexagonal pyramid 
hexagonal tiling  r{6,3}  rr{6,3}  tr{6,3} 
2uniform tilings  

