# Formulas and Tables

Elementary Geometry# Regular Polygon

Number of sides of a polygon: \(n\)

Interior angle: \(\alpha\)

Apothem: \(m\)

Area: \(S\)

Radius of the circumscribed circle: \(R\)

Perimeter: \(P\)

Semiperimeter: \(p\)

- A regular polygon is a convex polygon with equal sides and equal angles.

- 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. - Radius of the circumscribed circle

\(R = {\large\frac{a}{{2\sin \frac{\pi }{n}}}\normalsize}\) - 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. - Perimeter of a regular polygon

\(P = na\) - 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 \(p = {\large\frac{P}{2}\normalsize}\).