The purpose of this page is to state some facts about the angles of regular, in particular cyclic polygons. A rationale rather than a proof is given.
We consider n-sided polygons, where n is a positive integer, greater than 2.
In the diagram ABCD... is an n-sided polygon in a circle centre O. The sides of the polygon, AB, BC, etc are equal because it is a rgular polygon.
For AB, BC, CD, etc, the angles subtended at the centre are AOB, BOC, COD, etc, and they are all equal because triangles AOB, BOC, etc are congruent, (SSS) because two of the sides are equal to the radius of the circle and the other is the side of the polygon. Each angle is therefore radians, or degrees ■.
The internal angles of the polygon are ABC, BCD, etc. The angles ABO, OBC, etc are written as α, and the internal angles are 2α.
In each triangle, the angle at the centre is radians, and as the triangles OAB, OBC, etc are isosceles with OA, OB, etc as radii of the circle then 2α+=π, being the angle sum of the triangle. Hence, 2α=radians or degrees. Each internal angle is 2α, so each internal angle is radians or degrees ■.
The external angles are, for instance, CBX in the diagram, and are denoted by θ.
Because ABX is a straight line, 2α+θ=π.
So, as 2α=, then, θ= radians. So the external angles of the regular polygon are radians, or degrees ■.
Because there are n of these, there sum is 2π radians or 360° ■.