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In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a polygon with six edges and six vertices. The total of the internal angles of any hexagon is 720°.

A regular hexagon has Schläfli symbol {6} [1] and can also be constructed as a quasiregular truncated equilateral triangle, t{3}, which alternates two types of edges.


Regular hexagon
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.
The regular hexagon has a number of subsymmetries that can be seen by coloring or geometric variations

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 \( \tfrac{2\sqrt{3}}{3} \) 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 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 area of a regular hexagon of side length t is given by

\( A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2. \)

An alternative formula for the area is

\( A=\frac{3}{2}d \cdot t \)

where the length d is the distance between the parallel sides (also referred to as the flat-to-flat distance), or the height of the hexagon when it sits on one side as base, or the diameter of the inscribed circle.

Another alternative formula for the area if only the flat-to-flat distance, d, is known, is given by

\( A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866025404d^2.\)

The area can also be found by the formulas

A=ap/2

and

\( A\ =\ {2}a^2\sqrt{3}\ \simeq\ 3.464102 a^2,\)

where a is the apothem and p is the perimeter.

The perimeter of a regular hexagon of side length t is 6t, its maximal diameter 2t, and its minimal diameter \(\scriptstyle d\ =\ t\sqrt{3} \) .

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.


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 D3 symmetry.

A truncated hexagon, t{6} is an dodecagon, {12}, aternating 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.


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 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.


Related figures
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.[2]

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.[3]

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.[4]: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,[5]

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.[6]:Thm. 1
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[7]:p.184,#286.3 a principal diagonal d1 such that

\( \frac{d_1}{a} \leq 2 \)

and a principal diagonal d2 such that

\( \frac{d_2}{a} > \sqrt{3}. \)

Petrie polygons

The regular hexagon is the Petrie polygon for these regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:

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

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

References

Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
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
Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [4], Accessed 2012-04-17.
Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html

Inequalities proposed in “Crux Mathematicorum”, [5].

External links

Weisstein, Eric W., "Hexagon", MathWorld.
Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
Cymatics – Hexagonal shapes occurring within water sound images[dead link]
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)
supraHex A supra-hexagonal map for analysing high-dimensional omics data.

Mathematics Encyclopedia

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