Regular Polygon
GeometryA 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 ($3$ equal sides, $3$ equal angles) and a square ($4$ equal sides, $4$ right angles) are regular polygons.
Example
A stop sign is a regular octagon: all $8$ sides equal and all $8$ angles equal. A honeybee's honeycomb cell is a regular hexagon. A soccer ball uses regular pentagons and hexagons.
Key Insight
Regular polygons are the most symmetrical polygons. The more sides a regular polygon has, the more it looks like a circle. In fact, a circle can be thought of as a regular polygon with infinitely many sides.
Definition
A regular polygon is both equilateral (all sides equal) and equiangular (all angles equal). For an $n$-sided regular polygon: each interior angle $= (n-2) \times 180/n$ degrees, each exterior angle $= 360/n$ degrees. Regular polygons have $n$ lines of symmetry and $n$-fold rotational symmetry.
Example
Regular hexagon: interior angle $= (6-2) \times 180/6 = 120^\circ$. Exterior angle $= 360/6 = 60^\circ$. Lines of symmetry: $6$. Regular polygons that tile the plane: equilateral triangle ($60^\circ$ angles, $6$ fit at a point), square ($90^\circ$, $4$ fit), regular hexagon ($120^\circ$, $3$ fit).
Key Insight
Only three regular polygons tile the plane alone: equilateral triangle, square, and regular hexagon. The reason: the interior angles must divide evenly into $360^\circ$ at each vertex. $60\times6=360$, $90\times4=360$, $120\times3=360$. Pentagon angles ($108^\circ$) do not divide into $360$ evenly.
Definition
A regular $n$-gon has vertices at $\{e^{2\pi i k/n} : k = 0, 1, \ldots, n-1\}$ on the unit circle in the complex plane ($n$-th roots of unity). Its symmetry group is the dihedral group $D_n$ of order $2n$. The area of a regular $n$-gon with side length $s$ is $(ns^2)/(4\tan(\pi/n))$. The regular $n$-gon is constructible with compass and straightedge iff $n$ is a product of a power of $2$ and distinct Fermat primes (Gauss-Wantzel theorem).
Example
Regular pentagon ($n=5$): interior angle $= 108^\circ$, area $= (5s^2)/(4\tan(36^\circ)) = (5s^2\sqrt{5+2\sqrt{5}})/4$. Constructible since $5$ is a Fermat prime ($5 = 2^2 + 1$). Regular $7$-gon: NOT constructible ($7$ is not a Fermat prime).
Key Insight
Gauss proved at age $18$ that the regular $17$-gon is constructible ($17 = 2^4 + 1$ is a Fermat prime). This unexpected result linked number theory (properties of prime numbers) to geometry (constructibility). The connection between Fermat primes and constructible polygons remains one of the most striking results bridging algebra and geometry.