Ken Ward's Mathematics Pages

Geometry: Angles of a Regular Polygon

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.

• The sum of the external angles (or supplementary angles) is 2π radians or 360°.
• Each external angle is, therefore, radians, or  degrees.
• The angle subtended at the centre by the chords or sides is radians, or  degrees.
• The internal angles are each radians or degrees.

Page Contents

1. Angles Subtended at the centre
2. Internal Angles
3. External Angles

Comment

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.

Angles Subtended at the centre

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 ■.

Internal Angles

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 ■.

External Angles

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° ■.

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