Construction of Regular Polygons
Definition: A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure. Construction refers to drawing these shapes using only a compass and straightedge (unmarked ruler).
Key Properties of Regular Polygons
For a regular polygon with $n$ sides:
- Interior Angle: Each interior angle measures $$\theta = \frac{(n-2) \times 180°}{n}$$
- Exterior Angle: Each exterior angle measures $$\alpha = \frac{360°}{n}$$
- Central Angle: The angle subtended at the center by each side is $$\beta = \frac{360°}{n}$$
- Number of Diagonals: $$d = \frac{n(n-3)}{2}$$
The Inscribed Circle Method
The most common approach to constructing regular polygons is to inscribe them in a circle. Since all vertices of a regular polygon lie on a circle (the circumscribed circle), we divide the circle into $n$ equal arcs, each subtending a central angle of $\frac{360°}{n}$.
Gauss's Theorem: A regular $n$-gon is constructible with compass and straightedge if and only if $n$ is of the form: $$n = 2^k \cdot p_1 \cdot p_2 \cdots p_m$$ where each $p_i$ is a distinct Fermat prime ($3, 5, 17, 257, 65537$). This means polygons with $n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, \ldots$ sides are constructible.