Hexagon
A hexagon is a polygon with six sides and six interior angles summing to 720 degrees.
Formula
\text{Interior angle sum} = 720^\circ; \text{each regular angle} = 120^\circ
Definition
A hexagon is a polygon with six sides and six corners, whose angles add up to $720^\circ$ ($=(6-2)\times180$); a regular hexagon has all six sides equal, each angle $120^\circ$, $9$ diagonals, $6$ lines of symmetry, and divides into $6$ equilateral triangles sharing the center, giving area $(3\sqrt{3}/2)s^2$. Its vertices sit at $\{e^{i\pi k/3}:k=0,\ldots,5\}$ ($6$th roots of unity), symmetry group $D_6$ of order $12$, and it is the unique regular polygon (besides the triangle and square) that tiles $\mathbb{R}^2$ by itself, a fact confirmed rigorously by Hales's 1999 proof of the honeycomb conjecture.
Example
Honeycomb cells, snowflake cross-sections, and many floor tiles are hexagons; a bolt head is hexagonal so a wrench can grip it. For side $s=2$: area $=6\sqrt{3}\approx10.39$, long diagonal (across) $=2s=4$, short diagonal (between opposite edges) $=s\sqrt{3}\approx3.46$. The $6$th roots of unity $\{1, \omega, \omega^2, -1, -\omega, -\omega^2\}$ (where $\omega=e^{i\pi/3}$) have minimal polynomial $\Phi_6(x)=x^2-x+1$, the cyclotomic polynomial linking the hexagon to Galois theory.
Key Insight
The regular hexagon is the most efficient shape for tiling a flat surface, using the least border length to enclose the most area, which is why honeybees use hexagons for honeycombs, they store the most honey with the least wax. Dividing the hexagon into $6$ equilateral triangles from the center reveals why its area formula works and shows its internal six-fold symmetry, and the Hales proof of the honeycomb conjecture confirmed mathematically what bees seemed to "know" evolutionarily: the regular hexagonal grid minimizes total perimeter among all equal-area partitions of the plane.