Question

a circle of radius 6 cm is inscribed in a 5 sided regular polygon, find the lenghth of one side of the pentagon

Answer 1

Theorem 1..

If  [a]  is the length of the side of a regular [ n]-sided polygon and [ R ] is the radius of the circumscribed circle,  then

a/2 = R.sin[pi/n, or   a = 2R.sin[pi/n]

Use the formula of the  Theorem 1.  For the side length of the regular pentagon you have

a = 2R.sin[pi/5] = 10*sin(36°) = 10*0.5878 = 5.878 cm   (approximately).

The perimeter is five times the side length: P = 5*5.878 = 29.389 cm.

Answer. The side length is  5.878 cm. (approximately).