Hexagon geometry: area, apothem and diagonals
How the side of a regular hexagon fixes its area, apothem, circumradius and two diagonals, with the exact ratios behind each value.
Six equilateral triangles in disguise
A regular hexagon splits cleanly into six identical equilateral triangles that all meet at the centre. Each triangle has the same side length as the hexagon, and its height is the hexagon's apothem. Because the area of one equilateral triangle of side s is (square root of 3 over 4) times s squared, six of them sum to (3 times the square root of 3 over 2) times s squared. This is why the hexagon area constant of about 2.598076 appears: it is just six quarters of the square root of 3. Seeing the shape as six triangles also explains why the distance from the centre to a corner equals the side itself.
Apothem versus circumradius
Two radii describe any regular polygon. The circumradius reaches from the centre to a vertex, and for a hexagon it equals the side length exactly, which is a special feature of the six sided case. The apothem reaches from the centre to the midpoint of an edge and equals the square root of 3 over 2 times the side, roughly 0.866025 of it. The apothem is always the shorter of the two, and the ratio between them, about 0.866, is the cosine of 30 degrees. Knowing either radius lets you rebuild the side and therefore every other measurement.
The two diagonals
A hexagon has diagonals of two distinct lengths. The long diagonal connects opposite vertices, runs through the centre, and measures exactly twice the side, so it is also the diameter of the circle that passes through all six corners. The short diagonal connects vertices that are two apart, skipping one corner, and measures the square root of 3 times the side, about 1.732051 of it. There are three long diagonals and six short ones, giving nine diagonals in total, which matches the general polygon count of n times (n minus 3) over 2 for six sides.
Why hexagons tile so well
Interior angles of a regular hexagon are 120 degrees, and three of them meet snugly at any shared corner to make a full 360 degrees. That perfect fit lets hexagons cover a plane with no gaps or overlaps, one of only three regular polygons that can. For a fixed area a hexagonal cell also has a shorter perimeter than squares or triangles, so it minimises the wall material needed, which is the efficiency that honeybees, graphene, and chicken wire all exploit.