Regular Polygon
A regular polygon has all sides equal in length and all interior angles equal in measure.
Formula
\text{Each interior angle} = \frac{(n-2) \times 180^\circ}{n}
Definition
A regular polygon has all sides the same length AND all angles the same size; an equilateral triangle and a square are both regular polygons. For an $n$-sided regular polygon: each interior angle $=(n-2)\times180/n$ degrees, each exterior angle $=360/n$ degrees, and it has $n$ lines of symmetry and $n$-fold rotational symmetry. Its vertices sit at $\{e^{2\pi i k/n} : k=0,\ldots,n-1\}$ on the unit circle ($n$-th roots of unity), its symmetry group is the dihedral group $D_n$ of order $2n$, its area is $(ns^2)/(4\tan(\pi/n))$ for side length $s$, and it is constructible with compass and straightedge if and only if $n$ is a product of a power of $2$ and distinct Fermat primes (the Gauss-Wantzel theorem).
Example
A stop sign is a regular octagon, all $8$ sides and angles equal; a honeybee's honeycomb cell is a regular hexagon. For a regular hexagon: interior angle $=(6-2)\times180/6=120^\circ$, exterior angle $=60^\circ$; the equilateral triangle, square, and regular hexagon are the only regular polygons that tile the plane alone, since $60\times6=90\times4=120\times3=360$. A regular pentagon has interior angle $108^\circ$ and area $(5s^2\sqrt{5+2\sqrt{5}})/4$, and is constructible since $5=2^2+1$ is a Fermat prime, while a regular $7$-gon is not, since $7$ is not a Fermat prime.
Key Insight
The more sides a regular polygon has, the more it looks like a circle, which can be thought of as a regular polygon with infinitely many sides. Gauss proved at age $18$ that the regular $17$-gon is constructible ($17=2^4+1$ is a Fermat prime), an unexpected link between number theory (properties of primes) and geometry (constructibility) that remains one of the most striking results bridging algebra and geometry.