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° ■.
Ken Ward's Mathematics Pages/a>