# Regular Polygon

Side of a regular polygon: $$a$$
Number of sides of a polygon: $$n$$
Interior angle: $$\alpha$$
Apothem: $$m$$
Area: $$S$$
Radius of the inscribed circle: $$r$$
Radius of the circumscribed circle: $$R$$
Perimeter: $$P$$
Semiperimeter: $$p$$
1. A regular polygon is a convex polygon with equal sides and equal angles.
2. All interior angles in a regular polygon are equal and determined by the expression
$$\alpha = {\large\frac{{n – 2}}{n}\normalsize} \cdot 180^\circ,$$
where $$n$$ is the number of sides of the polygon.
3. Radius of the circumscribed circle
$$R = {\large\frac{a}{{2\sin \frac{\pi }{n}}}\normalsize}$$
4. The radius of the inscribed circle of a regular polygon coincides with the apothem (the perpendicular drawn from the centre to any side) and is given by the formula
$$r = m = {\large\frac{a}{{2\tan \frac{\pi }{n}}}\normalsize} =$$ $$\sqrt {{R^2} – {\large\frac{{{a^2}}}{4}}\normalsize},$$
where $$r$$ is the radius of the inscribed circle, $$m$$ is the apothem, $$R$$ is the radius of the circumscribed circle, $$a$$ is the side of the polygon.
5. Perimeter of a regular polygon
$$P = na$$
6. Area of a regular polygon
$$S = {\large\frac{{n{R^2}}}{2}\normalsize}\sin {\large\frac{{2\pi }}{n}\normalsize}$$
$$S = pr =$$ $$p\sqrt {{R^2} – {\large\frac{{{a^2}}}{4}}\normalsize}$$, where $$p = {\large\frac{P}{2}\normalsize}$$.