The value of 'Pi'

[shyamal kumar das]

THE VALUE OF Pi
Author: Shyamal Kumar Das (User ID: 642576)
E-mail ID: das.sh...@gmail.com
TO: All Mathematicians

On the Pi Day, I would like to state that I have calculated out the value of Pi in a different way.
In doing so, I have taken the help of my previous deduction which reveals that when the angle subtended at the center tends to zero, the difference between the arc length and the chord length tends to zero. The proof of the above statement is as follows:
Proof: Let us consider an angle of 4.5 degree arbitrarily.
Here, the angle subtended at the center =4.5 degree = Pi/40 radian.
The difference between arc length and chord length =
Pi/40 × R - 2 R × sin 2.25 = R (0.07854 -2× 0.03926) = R ( 0.07854 - 0.07852) =
0.00002 R, which means 2/ 100000 parts of unit radius.
Hence, it is revealed that when angle subtended at the center by an arc is 4.5 degree, the chord length is almost equal to the arc length. If we take 2.25 deg. or 1.125 deg. the difference will be lesser and least.

Now, with the help of the above deduction, I am going to find out the value of Pi.

Calculation: Let us consider a circle of unit radius and we will take half of it, i.e.,
semi-circle which subtends an angle of 180 deg. at the center. Now, we will bisect the given angle 8 times, to get a sector which is 1 / 256 part of the semi-circle and which subtends an angle of 180 / 256 or 45 / 64 deg. at the center.
If we join the two end points of the base of this sector, we will get an isosceles triangle. Hence, the given semi-circle consists of 256 nos. of sectors and so 256 nos. of isosceles triangles, considering, the arc length and chord length same.
The length of each base of these 256 triangles = 2 sin (45 / 128)
Hence, total length of bases of 256 triangles
= The semi-circumference of the circle of unit radius
Since, Pi is equal to semi-circumference of a circle of unit radius,
Pi = 256 × 2 sin 45/128 deg. = 512 sin 180/512 deg.

Again, area of semi-circle = area of one isosceles triangle × 256
= 1/2 × [2 sin ( 45 / 128) × cos ( 45 / 128)] × 256
= 1/2 × sin (45/ 64) × 256
So, area of the circle = 2× 1/2 sin (45/ 64) × 256
= sin (45/ 64) × 256 = 256 sin (180 / 256)
Since, Pi is equal to area of a circle of unit radius,
Pi = 256 sin (180 / 256)
The above calculation is made taking 8 times bisections of a circle, which means 256 equal parts of the semi-circle and/or 512 equal parts of the whole circle.
If we take 9 times bisections of the circle, which means 512 equal parts of the semi-circle and/or 1024 equal parts of the whole circle,
Pi = 1024 sin 180/1024 deg. or 512 sin 180/512 deg.
(Here, 2 to the power 9 = 512)
>From the above, we can derive a new formula of Pi, which is as below:
The value of Pi = X sin 180/X deg., where X = No. of equal parts of the circle .
Here, please note, for bigger circle, no. of bisections will be more. So, X depends on the magnitude of the radius of the given circle.