Hexagon

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Regular hexagon
Hexagon.png
A regular hexagon, {6}
Edges and vertices 6
Schläfli symbols {6}
t{3}
Coxeter–Dynkin diagrams CDW ring.pngCDW 6.pngCDW dot.png
Image:CDW_ring.pngCDW 3.pngImage:CDW_ring.png
Symmetry group Dihedral (D6)
Area
(with t=edge length)
A = \frac{3 \sqrt{3}}{2}t^2
 \simeq 2.598076211 t^2.
Internal angle
(degrees)
120°

In geometry, a hexagon is a polygon with six edges and six vertices. A regular hexagon has the Schläfli symbol {6}.

Contents

Regular hexagon

A regular hexagon is constructible with compass and straightedge. Click image to see a step-by-step animated method of this, given by Euclid's Elements, Book IV, Proposition 15.

The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120° and the hexagon has 720 degrees. It has six lines of symmetry. 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.

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.

The perimeter of a regular hexagon of side length t\,\! is, of course, 6t\,\!, its maximal diameter 2t\,\!, and its minimal diameter t\sqrt{3}\,\!.

There is no platonic solid made of regular hexagons. 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.

Hexagons: natural and artificial

See also

References

  • Arnone, Wendy. 2001. Geometry for Dummies. Hoboken, NJ: For Dummies (Wiley). ISBN 0764553240
  • Hartshorne, Robin. 2002. Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. New York, NY: Springer. ISBN 0387986502
  • Leff, Lawrence S. 1997. Geometry the Easy Way. Hauppauge, NY: Barron’s Educational Series. ISBN 0764101102
  • Stillwell, John. 2005. The Four Pillars of Geometry. Undergraduate Texts in Mathematics. New York, NY: Springer. ISBN 0387255303

External links

All links retrieved June 19, 2013.


Polygons
Triangle • Quadrilateral • Pentagon • Hexagon • Heptagon • Octagon • Enneagon (Nonagon) • Decagon • Hendecagon • Dodecagon • Triskaidecagon • Pentadecagon • Hexadecagon • Heptadecagon • Enneadecagon • Icosagon • Chiliagon • Myriagon

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