Hexagon
GeometryA 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. The six angles inside add up to $720^\circ$. A regular hexagon has all six sides equal and all angles measuring $120^\circ$.
Example
Honeycomb cells, snowflake cross-sections, and many floor tiles are hexagons. A stop sign has $8$ sides (octagon), but a typical bolt head has a hexagonal shape so that a wrench can grip it.
Key Insight
The regular hexagon is the most efficient shape for tiling a flat surface - it uses the least border length to enclose the most area. This is why honeybees use hexagons for their honeycombs: it is the shape that stores the most honey with the least wax.
Definition
A hexagon has $6$ sides, $6$ vertices, interior angle sum $= (6-2) \times 180 = 720^\circ$. A regular hexagon has interior angles of $120^\circ$ and can be divided into $6$ equilateral triangles, all sharing the center. It has $9$ diagonals and $6$ lines of symmetry. The regular hexagon tiles the plane.
Example
Regular hexagon with side $s$: area $= (3\sqrt{3}/2)s^2$. For $s = 2$: area $= 6\sqrt{3}$ approximately $10.39$. The long diagonal (across) $= 2s = 4$. The short diagonal (between opposite edges) $= s\sqrt{3}$ approximately $3.46$.
Key Insight
Dividing a regular hexagon from the center into $6$ equilateral triangles reveals why the area formula works: $6$ triangles each with area $(\sqrt{3}/4)s^2$, totaling $(3\sqrt{3}/2)s^2$. This decomposition also shows the hexagon's internal six-fold symmetry.
Definition
A regular hexagon has vertices at $\{e^{i\pi k/3} : k=0,\ldots,5\}$ on the unit circle, forming the $6$th roots of unity. Its symmetry group is $D_6$ (dihedral, order $12$). It is the unique regular polygon (other than triangle and square) that tiles $\mathbb{R}^2$ by itself. By the honeycomb conjecture (proved by Hales, 1999), the regular hexagonal tiling is the most efficient partition of the plane into equal-area regions.
Example
The Hales proof of the honeycomb conjecture confirmed mathematically what bees "knew" evolutionarily: among all ways to divide the plane into equal areas, the regular hexagonal grid minimizes total perimeter. This connects the hexagon to the isoperimetric problem in a discrete/periodic setting.
Key Insight
The $6$th roots of unity (hexagon vertices) are $\{1, \omega, \omega^2, -1, -\omega, -\omega^2\}$ where $\omega = e^{i\pi/3}$. These roots connect the hexagon to cyclotomic fields in algebraic number theory. The minimal polynomial of $\omega$ over $\mathbb{Q}$ is the cyclotomic polynomial $\Phi_6(x) = x^2 - x + 1$, linking the hexagon to Galois theory.