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For other uses, see Hexagon (disambiguation).
Regular hexagon
Regular polygon 6 annotated.svg
A regular hexagon
Type Regular polygon
Edges and vertices 6
Schläfli symbol {6}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.png
Symmetry group D6, 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 polygon with six edges and six vertices. A regular hexagon has Schläfli symbol {6}. The total of the internal angles of any hexagon is 720°.

Hexagonal structures[edit]

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.

Regular hexagon[edit]

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 has all sides of the same length, and that length is also the radius of the circumscribed 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.

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 area is A = 1.5dt 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 \scriptstyle 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.

Tesselations by hexagons[edit]

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.

Cyclic hexagon[edit]

A cyclic hexagon is any hexagon inscribed in a circle. If the successive sides of the 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.[1]

Hexagon inscribed in a conic section[edit]

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.

Hexagon tangential to a conic section[edit]

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,[2]


Convex equilateral hexagon[edit]

The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[3]: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}.

Related figures[edit]

Truncated triangle.svg
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. This form only has D3 symmetry. In this figure, the remaining edges of the original triangle are drawn blue, and new edges from the truncation are red.
The hexagram can be created as a stellation process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices.
Medial triambic icosahedron face.png
A concave hexagon
Great triambic icosahedron face.png
A self-intersecting hexagon (star polygon)
Cube petrie polygon sideview.png
A (nonplanar) skew regular hexagon, within the edges of a cube

Petrie polygons[edit]

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

(3D) (5D)
Cube petrie.png
Octahedron petrie.png
5-simplex t0.svg
5-simplex t1.svg
Rectified 5-simplex
5-simplex t2.svg
Birectified 5-simplex

Polyhedra with hexagons[edit]

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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel p.pngCDel node.png and CDel node 1.pngCDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.png.

Archimedean solids
Tetrahedral Octahedral Icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Truncated tetrahedron.png
truncated tetrahedron
Truncated octahedron.png
truncated octahedron
Great rhombicuboctahedron.png
truncated cuboctahedron
Truncated icosahedron.png
truncated icosahedron
Great rhombicosidodecahedron.png
truncated icosidodecahedron

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

Tetrahedral Octahedral Icosahedral
Alternate truncated cube.png
Chamfered tetrahedron
Truncated rhombic dodecahedron2.png
Chamfered cube
Truncated rhombic triacontahedron.png
Chamfered dodecahedron

There are also 9 Johnson solids with regular hexagons:

Triangular cupola.png
triangular cupola
Elongated triangular cupola.png
elongated triangular cupola
Gyroelongated triangular cupola.png
gyroelongated triangular cupola
Augmented hexagonal prism.png
augmented hexagonal prism
Parabiaugmented hexagonal prism.png
parabiaugmented hexagonal prism
Metabiaugmented hexagonal prism.png
metabiaugmented hexagonal prism
Triaugmented hexagonal prism.png
triaugmented hexagonal prism
Augmented truncated tetrahedron.png
augmented truncated tetrahedron
Triangular hebesphenorotunda.png
triangular hebesphenorotunda
Hexagonal prism.png
Hexagonal prism
Hexagonal antiprism.png
Hexagonal antiprism
Hexagonal pyramid.png
Hexagonal pyramid
Other symmetric polyhedral with hexagons
Truncated triakis tetrahedron.png
Truncated triakis tetrahedron
Hexpenttri near-miss Johnson solid.png

Regular and uniform tilings with hexagons[edit]

Uniform tiling 63-t0.png
The hexagon can form a regular tessellate the plane with a Schläfli symbol {6,3}, having 3 hexagons around every vertex.
Uniform tiling 63-t12.png
A second hexagonal tessellation of the plane can be formed as a truncated triangular tiling or rhombille tiling, with one of three hexagons colored differently.
Uniform tiling 333-t012.png
A third tessellation of the plane can be formed with three colored hexagons around every vertex.
Isohedral tiling p6-7.png
The hexagonal tiling can be distorted, like these centrosymmetric hexagons
Uniform tiling 63-t1.png
Trihexagonal tiling
Uniform tiling 333-t01.png
Trihexagonal tiling
Uniform polyhedron-63-t02.png
Rhombitrihexagonal tiling
Uniform polyhedron-63-t012.png
Truncated trihexagonal tiling

Hexagons: natural and human-made[edit]

See also[edit]


  1. ^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
  2. ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [5], Accessed 2012-04-17.
  3. ^ Inequalities proposed in “Crux Mathematicorum”, [6].

External links[edit]