Hexagon
From Free net encyclopedia
- For other uses, see Hexagon (disambiguation).
In geometry, a hexagon is a polygon with six edges and six vertices. Its Schläfli symbol is {6}.
The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120°. 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 <math>a\,\!</math> is given by
<math>A = \frac{3}{2}a^2 \cot \frac{\pi}{6} = \frac{3 \sqrt{3}}{2}a^2 \simeq 2.59808 a^2.</math>
The circumference of a hexagon of side length <math>a\,\!</math> is, of course, <math>6a\,\!</math>, its maximal diameter <math>2a\,\!</math>, and its minimal diameter <math>\sqrt{3}a\,\!</math>.
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.
In French, the term l'hexagone (the hexagon) is often used as a nickname for France, since the country is shaped somewhat like a hexagon.
Hexagon construction
A regular hexagon is constructible with compass and straightedge. The following is a step-by-step animated method of this, given by Euclid's Elements, Book IV, Proposition 15. Image:HexagonConstructionAni.gif
See also
nl:Zeshoek pl:Sześciokąt pt:Hexágono