The Regular Polygon
Rhombus is a quadrilateral with all sides equal (equilateral). Rectangle is a quadrilateral with all included angles are equal (equiangular). Square is both equilateral and equiangular, thus square is a regular polygon. Regular polygons are polygons with all sides equal and all included angles equal. Meaning, regular polygons are both equilateral and equiangular.
Properties of regular polygons
- The center of the circumscribing circle, the center of inscribed circle, and the center of polygon itself are coincidence.
- All sides of regular polygon are equal in length; it is denoted by x in the figure.
- All included angles are equal; it is denoted by β.
- All external angles α, are equal.
- Central angles of each segment are equal; it is denoted by θ.
- The apothem is the radius of the inscribed circle, r.
- The number of sides is equal to the number of vertices, both are denoted by n.
- Diagonals of regular polygon will cross each other at the center. Length of diagonals is equal to the diameter of the circumscribing circle.
- Segment of regular polygon is an isosceles triangle whose equal sides are radius of circumscribing circle; its area is denoted by A1.
Formulas for a Regular Polygon
A1 = area of one segment
A = total area
x = length of side
r = radius of the inscribed circle (apothem)
R = radius of the circumscribing circle
n = number of sides
θ = central angle
α = exterior angle
β = interior angle