Hexagon

A hexagon is a polygon with six straight sides and six corners (angles). It is one of the most frequently taught shapes in basic geometry, and is used in architecture, design, and mathematical analysis.

The sum of the interior angles of any hexagon is 720 degrees. This is derived from the general formula (n − 2) × 180, where n is the number of sides. Hexagons fall into two main types: regular and irregular.

Regular hexagon

A regular hexagon has six equal sides and six angles, each measuring 120 degrees. It has six lines of symmetry and rotational symmetry of order six. It can be divided into six equilateral triangles, making it possible to construct one easily using only a compass and straightedge.

If each side has length a, the area of a regular hexagon is (3√3 ⁄ 2) × a². The perimeter is 6a. The distance from the center to each corner equals the length of a side, a property unique to the hexagon.

Tiling of plane space

The regular hexagon is one of only three regular polygons—the triangle, square, and hexagon—that can fill a flat plane with no gaps or overlaps. This pattern is called tessellation. Because three angles of 120 degrees sum to 360 degrees, three hexagons meet at each vertex.

Natural occurrence and application

Hexagonal form appears in nature: honeycomb cells are hexagonal units with shared walls, which minimizes the wax required compared to the area enclosed. Similarly, the structure of snowflake crystals and certain atomic arrangements in materials (for example, the carbon ring in benzene and layers in graphite) have hexagonal form.

In engineering, hexagons are used in the heads of bolts and nuts, which allows a tool to grip from six different angles. They are also used in the units of networks and in hexagonal grids for strategy games, since each unit has straight contact with six neighboring units at equal distance.