Chapter 12.5, Problem 23WE

To determine

Explain why the number

  $2\pi$ lies between the perimeter of the regular polygon that is inscribed in a unit circle and the perimeter of the regular polygon circumscribed about a unit circle.

Explanation of Solution

Calculation:

Since the unit circle lies between the regular polygon inscribed in the circle and the regular polygon circumscribed about the circle, so , its perimeter will also lie between the perimeter of the regular polygon inscribed in the unit circle and the perimeter of the regular polygon circumscribed about the unit circle.

Hence,

perimeter of the regular polygon inscribed in the unit circle < perimeter of the unit circle < perimeter of the regular polygon circumscribed about the unit circle.

perimeter of the regular polygon inscribed in the unit circle < $2\pi (1)=2\pi$< perimeter of the regular polygon circumscribed about the unit circle.

Hence explained.