A regular hexagon has Schläfli symbol {6}[2] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.

The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side.

The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials.

The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: The area of a regular hexagon For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD.

The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern.

The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern.

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.

This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces.

Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation.

In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation.

Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.

If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.

Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.

In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.

The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.

The prefix "hex-" originates from the Greek word "hex," meaning six, while "sex-" comes from the Latin "sex," also signifying six.

Some linguists and mathematicians argue that since many English mathematical terms derive from Latin, the use of "sexagon" would align with this tradition.

Historical discussions date back to the 19th century, when mathematicians began to standardize terminology in geometry.

However, the term "hexagon" has prevailed in common usage and academic literature, solidifying its place in mathematical terminology despite the historical argument for "sexagon."

The consensus remains that "hexagon" is the appropriate term, reflecting its Greek origins and established usage in mathematics.