Re: The value of 'Pi'

[Anyssa]

> 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
> e = Pi/40 radian.
> The difference between arc length and chord length =
> Pi/40 × R - 2 R × sin 2.25 = R (0.07854 -2×
> 2× 0.03926) = R ( 0.07854 - 0.07852) =
> 0.00002 R, which means 2/ 100000 parts of unit
> it radius.
> Hence, it is revealed that when angle subtended at
> t 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
> is 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
> 2 sin (45 / 128)
> Hence, total length of bases of 256 triangles
> = The semi-circumference of the circle of unit
> nit 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
> 80/512 deg.
>
> Again, area of semi-circle = area of one isosceles
> triangle × 256
> = 1/2 × [2 sin ( 45 / 128) × cos ( 45 / 128)]
> / 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 sin (180 / 256)
> Since, Pi is equal to area of a circle of unit
> nit 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.

I better understand now. Pi is a very interesting value.