# Hexagon

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In geometry, a hexagon is a polygon with six edges and six vertices. A regular hexagon has Schläfli symbol {6}. The total of the internal angles of any hexagon is 720 degrees.

## Contents

### Regular hexagon

A regular hexagon has all sides of the same length, and all internal angles are 120°. 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. 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 $A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.$

An alternative formula for area is $\ A = 1.5dt$

Length d is the distance between the parallel sides, or the height of the hexagon when it sits on one side as base, or the (minimal) diameter of the inscribed circle.

The area can also be found by the formula A=ap/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 $t\sqrt{3}\,\!$.