Area of a circle inscribe a hexagon. Cut the hexagon into six triangles by splitting it from the center. Two opposite triangles both touch two common diameters; slide them along one so the radial edges are adjacent. They now form a parallelogram, with the hexagon sides making two opposite edges, one of which is the base, "s". Two radial edges form slanted sides, and the height, "h" is equal to its apothem (as in the Archimedes proof). In fact, we can assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase
Hexagon formula_7 of its circumscribed circle. If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C, then . The "regular hexagon" has Dih symmetry, order 12. There are 3 dihedral subgroups: Dih, Dih, and Dih, and 4 cyclic subgroups: Z, Z, Z, and Z. These symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long