Chapter 12.5, Problem 24WE

(a)

To determine

Find the perimeter of a regular polygon having n sides inscribed in a unit circle and perimeter of a regular polygon having n sides circumscribed about the unit circle.

Answer to Problem 24WE

   ,

Explanation of Solution

Formula Used:

Law of Sines:

  

  

Calculation:

The n -gon has n sides and if we join the sides to the centre of the circle, it forms n isosceles triangles with equal sides length 1 .

  

Since there are n equal triangles, the angle to the centre of each triangle is .

Let base of each triangle is x .

In the isosceles triangle , the central angle is , the other two angles are equal since the triangle is isosceles , so each angle is .

Use the law of sines to find x :

  

Since the n -gon has n sides , and one of the side is x , so , the perimeter of the n -gon inscribed inside a unit circle is nx :

  

The n -gon circumscribed about the unit circle has n sides and if we join the sides to the centre of the circle, it forms n triangles:

  

Since there are n equal triangles, the angle to the centre of each triangle is .

So, the central angle of the right angled triangle that bisects the triangle is

Let base of each right angled triangle is x and height is 1.

So,

  

Eace side of the n - gon is 2x =

Since the n -gon has n sides , and one of the side is , so , the perimeter of the n -gon circumscribed about a unit circle is:

  

(b)

To determine

Find where the number is approached when n gets larger and larger:

   ,

Answer to Problem 24WE

both the numbers approaches 0 as n becomes larger and larger.

Explanation of Solution

Calculation :

As n becomes larger and larger , becomes smaller and smaller. So,

  

Hence , both the numbers approaches 0 as n becomes larger and larger.