# Hexagon

Last updated
Regular hexagon
A regular hexagon
Type Regular polygon
Edges and vertices 6
Schläfli symbol {6}, t{3}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D6), order 2×6
Internal angle (degrees)120°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "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°.

## Regular hexagon

A regular hexagon has Schläfli symbol {6} [1] 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, drawing a circular arc from point A and point B gives the intersection M, 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 or circumcircle, which equals ${\displaystyle {\tfrac {2}{\sqrt {3}}}}$ 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 D6. 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:

${\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}$  and, similarly, ${\displaystyle d={\frac {\sqrt {3}}{2}}D.}$

The area of a regular hexagon

{\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{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${\displaystyle {}=6R=4r{\sqrt {3}}}$, so

{\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}

The regular hexagon fills the fraction ${\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}$ 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 circumcircle between B and C, then PE + PF = PA + PB + PC + PD.

It follows from the ratio of circumradius to inradius that the height-to-width 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.

### Point in plane

For an arbitrary point in the plane of a regular hexagon with circumradius ${\displaystyle R}$, whose distances to the centroid of the regular hexagon and its six vertices are ${\displaystyle L}$ and ${\displaystyle d_{i}}$ respectively, we have [2]

${\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),}$
${\displaystyle d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),}$
${\displaystyle d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).}$

If ${\displaystyle d_{i}}$ are the distances from the vertices of a regular hexagon to any point on its circumcircle, then [2]

${\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}$

## Symmetry

The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)

These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. [3] 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 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
Dih6Dih2Z2Dih1Z1

### 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 projection12 rhomb dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 12m(m − 1) parallelograms. [4] 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
2DRhombsParallelograms
Regular {6}Hexagonal parallelogons
3DSquare facesRectangular 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 D3 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.

### Self-crossing hexagons

There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:

Self-intersecting hexagons with regular vertices
Dih2Dih1Dih3

Figure-eight

Center-flip

Unicursal

Fish-tail

Double-tail

Triple-tail

## Hexagonal structures

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 3-space by translation.

Hexagonal prism tessellations
Form Hexagonal tiling Hexagonal prismatic honeycomb
Regular
Parallelogonal

## Tesselations by hexagons

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. [5]

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. [6]

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. [7] :p. 179

## 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, [8]

${\displaystyle a+c+e=b+d+f.}$

## 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. [9] :Thm. 1

## Skew hexagon

A skew hexagon is a skew polygon with six 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 three 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 D3d, [2+,6] symmetry, order 12.

The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.

 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:

4D5D

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 [10] :p.184,#286.3 a principal diagonal d1 such that

${\displaystyle {\frac {d_{1}}{a}}\leq 2}$

and a principal diagonal d2 such that

${\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}$

### 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 .

There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):

There are also 9 Johnson solids with regular hexagons:

## Related Research Articles

A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.

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 vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

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 eight-sided polygon or 8-gon.

In geometry, a decagon is a ten-sided polygon or 10-gon. 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 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

In geometry, a dodecagon or 12-gon is any twelve-sided 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 equal-length 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, the triaugmented triangular prism is a convex polyhedron with 14 equilateral triangles as its faces. Also called the tetracaidecadeltahedron, tetrakaidecadeltahedron, tetrakis triangular prism, or tricapped trigonal prism, it is an example of a deltahedron and a Johnson solid. It can be constructed by attaching equilateral square pyramids to each of the three square faces of a triangular prism.

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided 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 sixteen-sided polygon.

In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry, an icositetragon or 24-gon is a twenty-four-sided 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.

## References

1. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN   9780521098595, archived from the original on 2016-01-02, retrieved 2015-11-06.
2. Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:.
3. 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)
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
5. Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
6. Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum . 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17.
7. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
8. Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", Archived 2012-05-11 at the Wayback Machine , Accessed 2012-04-17.
9. Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12.
10. Inequalities proposed in "Crux Mathematicorum", Archived 2017-08-30 at the Wayback Machine .
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds