Hexagon

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Hexagon

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a hexagon is a polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
with six edges and six vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
. A regular hexagon has Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
.

internal angle

Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s of a regular hexagon (one where all sides and all angles are equal) are all 120°

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
and the hexagon has 720 degrees T.

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Encyclopedia

In geometry

Geometry

Polygon

Vertex (geometry)

Schläfli symbol

Regular hexagon

The internal angle

Angle

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
and the hexagon has 720 degrees T. It has 6 rotational symmetries

Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
and 6 reflection symmetries

Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
, making up the dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
D₆. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice its sides in length. Like square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s and equilateral

Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths....
triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellation

Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
s. The cells of a beehive

Beehive (beekeeping)

A beehive is an enclosed structure in which some honey bee species of the genus Apis live and raise their young. Natural beehives are naturally-occurring structures occupied by honey bee colonies, while domesticated honey bees live in man-made beehives, often in an apiary....
honeycomb

Honeycomb

A honeycomb is a mass of hexagonal waxcells built by honey bees in their beehive to contain their larva and stores of honey and pollen.beekeeping may remove the entire honeycomb to harvest honey....
are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram

Voronoi diagram

In mathematics, a Voronoi diagram, named after Georgy Voronoy, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation , is a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points....
of a regular triangular lattice is the honeycomb tessellation of hexagons.

The area of a regular hexagon of side length is given by

Also, it can be found by the formula A=ap/2, where a is the apothem

Apothem

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides....
and p is the perimeter.

The perimeter of a regular hexagon of side length is, of course, , its maximal diameter , and its minimal diameter .

There is no platonic solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
made of regular hexagons. The archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s with some hexagonal faces are the truncated tetrahedron

Truncated tetrahedron

The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....
, truncated octahedron

Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
, truncated icosahedron

Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
(of soccer ball and fullerene

Fullerene

Fullerene are a family of carbon Allotropy, molecules composed entirely of carbon, in the form of a hollow sphere, ellipsoid, cylinder , or plane....
fame), truncated cuboctahedron

Truncated cuboctahedron

The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....
and the truncated icosidodecahedron

Truncated icosidodecahedron

The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....
.

Related figures

A regular hexagon can also be created as a truncated Truncation (geometry) In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex.... equilateral triangle Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?.... , with Schläfli symbol t. 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 Hexagram A hexagram is a six-pointed geometric star figure, or 2, the compound of two equilateral triangle s. The intersection is a regular hexagon.While generally recognized as a symbol of Jewish identity it is used also in other historical, religious and cultural contexts, for example in #Use of the Star by Arabs and Muslims, and #Occurrence in... can be created as a stellation Stellation Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again.... process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices.

Hexagons: natural and human-made

External links

With interactive animation
- from Space.com
Space.com

Space.com is a space and astronomy news website. Its stories are often syndicated to other mass media outlets, including CNN, MSNBC, Yahoo!, and USA Today....
(27 March 2007)

Hexagon

Regular hexagon

Related figures

Hexagons: natural and human-made

See also

External links