Question

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

(A) There is a regular polygon with r/R = 1/√2

(B) There is a regular polygon with r/R = 2/3

(C) There is a regular polygon with r/R = √3/2

(D) There is a regular polygon with r/R = 1/2

Answer 1

Correct option  (B) There is a regular polygon with r/R = 2/3

Consider a regular polygon of n sides. Draw a line segment from its centre to each of its n sides to get n number of similar triangles which will look as shown in Fig.

Let θ be angle AOB. Therefore, 

θ = 360°/n

Then from trigonometry of right triangles, we have

So we have, 

r = a/2cot π/n

where a is the side of polygon. Therefore,