A hexagon is a six-sided polygon. A regular hexagon has six equal sides and six equal interior angles, each measuring 120°. Regular hexagons are highly symmetric and can be constructed from six equilateral triangles. They're commonly seen in nature (like honeycombs) and tiling patterns.
Total distance around the hexagon:
\[ P = 6a \]
The area of a regular hexagon can be calculated as:
\[ A = \frac{3\sqrt{3}}{2} a^2 \]
or, using the apothem \(r\):
\[ A = \frac{1}{2} P \times r = 3a \times r \]
\[ r = \frac{a\sqrt{3}}{2} \]
\[ R = a \]
The hexagon can be broken into 6 equilateral triangles. Hence:
\[ A = 6 \times \left( \frac{\sqrt{3}}{4} a^2 \right) = \frac{3\sqrt{3}}{2} a^2 \]