 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}