Side of a regular polygon: \(a\)
Number of sides of a polygon: \(n\)
Interior angle: \(\alpha\)
Apothem: \(m\)
Area: \(S\)
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: \(p\)
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 \(p = {\large\frac{P}{2}\normalsize}\).
