Formula for PI

The ancient mathematicians calculated the value of PI to a number of significant digits.  Everyone knows the definition  of PI: ratio of circumference of a circle to it’s diameter.

Now, how did the value 3.1415… come? How might have the first person calculated the numerical value of PI?

Key idea: We need a good approximation to the value of circumference. How do we do that? Take an infinitesimally small arc and approximate its length to that of the corresponding chord.

Consider an arc (AC) of the circle describing angle : theta (in degrees) at the centre of the circle (B)

.    A
.      /
r   /
. /   Angle B=theta
/——————-
B             r       C

Here, r is the radius.

We can complete the triangle by joining AC.

Now length of side AC =   sin(theta) * [ r / sin[(180-theta)/2] ] (Using the sine formula on triangle ABC, and since  angle A = angle C = (180 – theta) / 2])

Now we can use simple results to get :    length of side AC = 2 * r * sin(theta/2).

( since  : sin [(180 – theta)/2] = sin (90 – theta/2) = cos (theta/2)   &  sin(theta) = 2 sin(theta/2) cos(theta/2). )

Key idea (once again): As (theta->0),  length of side AC -> length of arc AC;
and circumference of circle = length of arc AC * (360/theta)

Hence, ratio of circumference to diameter = PI=  Lim (theta ->0) (360/ theta) * (length of arc AC) / (2 * r)

= Lim (theta ->0) (360/theta) * [2 * r * sin(theta/2)]  / [2*r]

= Lim (theta->0) 360 *  sin(theta/2) / theta.
So, there we have it: PI = Lim (theta->0) 360 *  sin(theta/2) / theta.

We can check using a calculator:

For theta =  1:    PI = 360 * sin(0.5) / 1.0  = 3.14155

For theta = 0.002:   PI = 360 * sin(0.001) / 0.002 = 3.14159.

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