hexagons

A hexagon is a polygon with six sides and six vertices. In Euclidean geometry, hexagons are typically simple (non-self-intersecting) and can be convex or concave. The sum of interior angles in any hexagon is 720 degrees, and a convex hexagon has nine diagonals. Hexagons can be classified into various subclasses, the best known being the regular hexagon, in which all sides and all interior angles are equal.

In a regular hexagon with side length s, the perimeter is 6s and the area is (3√3/2)·s². The circumradius equals s and the apothem equals (√3/2)·s, which yields an area formula also equal to (1/2)·P·apothem. The long diagonals, connecting opposite vertices, have length 2s, while the short diagonals, connecting vertices two apart, have length √3·s. The distance between opposite sides (the width) is √3·s.

Hexagons have notable tiling properties. Regular hexagons tessellate the plane in a honeycomb pattern without gaps or overlaps, illustrating a regular tessellation of the plane. The symmetry of a regular hexagon is described by the dihedral group D6, consisting of rotations by multiples of 60 degrees and reflections, for a total of 12 symmetries.

Hexagons appear in nature and human design. Beehives use hexagonal cells because the shape efficiently partitions space with minimal perimeter for a given area. The lattice arrangement of carbon atoms in graphene is a hexagonal tiling, and hexagonal patterns are common in tiling, woodworking, and board games, where hex grids offer uniform distance between adjacent cells.