How to derive the value of pi?

Answer 1

The perimeter of a inscribed regular polygon on a circle gets arbitrarily close to the circumference of the circle as the number of sides of the polygon approaches infinity.

So we can say that the limit of the perimeter of the polygon as the number of sides n approaches infinity is the circumference of the circle.

The perimeter of the polygon of n sides is equal to n*x(the length of one side).

By connecting the center of the circle to the vertices of the polygon, we form n number of triangles and by connecting the center to x/2, we form 2n number of right triangles with the angle adjacent to the center equal to 360/2n or 180/n and the hypotenuse equal to the radius.

By using the trigonometric functions, we can solve for x:

Sin(180/n) = (x/2)/r

r* Sin(180/n) = x/2

2r* Sin(180/n) = x

Then we can substitute this in the first equation for the perimeter of the regular polygon:

n*2r* Sin(180/n) = perimeter of the polygon = circumference of the circle if n is very large.

Substituting Diameter for 2r:

D*n*sin(180/n) = perimeter = circumference if n is very large

D*n*sin(180/n) = C

Getting the ratio:

C/D = n*sin(180/n)

pi = n*sin(180/n)