Elementary Geometry

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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. Regular polygon
  3. 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.
  4. Radius of the circumscribed circle
    \(R = {\large\frac{a}{{2\sin \frac{\pi }{n}}}\normalsize}\)
  5. 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.
  6. Perimeter of a regular polygon
    \(P = na\)
  7. 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 \(p = {\large\frac{P}{2}\normalsize}\).