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BodhiAI
Nov. 12, 2019

Regular Polygon:

Regular Polygon 

A. Radii of the Inscribed & Circumscribed Circles of a Regular Polygon 

A polygon is said to be a regular polygon if all its sides are equal and all its angles are equal. The circle passing through all the vertices of a regular polygon is called its circumscribed circle and the circle which touches all the sides of a regular polygon is called its inscribed circle.

Suppose AB, BC and CD are three consecutive sides of a regular polygon having total number of sides n. Draw the bisectors of the angles ABC and BCD meeting at O and draw OM perpendicular to BC. Obviously O is the centre of both the circumcircle and the in-circle of this polygon.

Let a is the length of side of the regular polygon.

Since the whole angle at the centre O is 2π radians, therefore angle subtended by each side is 

Now, OB = R = radius of the circum-circle and OM = r= radius of the in-circle. In right angled ΔOBM

we have 

Again, from ΔOBM we have 

B.   Area of a Regular Polygon

The area of the regular polygon shown in the above figure is n times the area of

The area of the polygon

{in the terms of the length of a side}

Also, the area of the polygon A = n. (BM.OM)

{in terms of radius of circum-circle}

Again, the area of the polygon A = n. (OM.BM)

{in terms of the radius of in-circle}