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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
CDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Dihedral (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. 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 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.


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


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\ =\ {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[edit]

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.

Regular polygon 6 annotated.svg Truncated triangle.svg Regular truncation 3 1000.svg Regular truncation 3 1.5.svg Regular truncation 3 0.55.svg Hexagram.svg Regular polygon 12 annotated.svg Regular polygon 3 annotated.svg
t{3} = {6}
Hypertruncated triangles Stellated
Star figure 2{3}
t{6} = {12}
h{6} = {3}
Medial triambic icosahedron face.png Great triambic icosahedron face.png 3-cube t0.svg Hexagonal cupola flat.png Cube petrie polygon sideview.png
A concave hexagon A self-intersecting hexagon (star polygon) Dissected {6} Extended
Central {6} in {12}
A skew hexagon, within cube

Related figures[edit]

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.

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.

Cyclic hexagon[edit]

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.[1]

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.[2]

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.[3]:p. 179

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,[4]


Convex equilateral hexagon[edit]

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[5]: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[edit]

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

3D 4D 5D
Cube petrie.png
Octahedron petrie.png
3-3 duoprism ortho-Dih3.png
3-3 duoprism
3-3 duopyramid ortho.png
3-3 duopyramid
5-simplex t0.svg

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. ^ 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
  3. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
  4. ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [4], Accessed 2012-04-17.
  5. ^ Inequalities proposed in “Crux Mathematicorum”, [5].

External links[edit]