Doglas, Y. (2006). A New Approximation for Pi (3.141592654). PHILICA.COM Observation number 17.
A New Approximation for Pi (3.141592654)

Yeo Doglasunconfirmed user (Singapore, Independent Researcher)

Published in matho.philica.com

Observation
It is easy to deduce from the Maclaurin’s series that when
x is small, sin x=x
.

Expanding from this idea,

one can get an approximation of PI using trigonometrical ratios.

Pi= N sin(180/N)

As N approaches infinity (ie. gets very large), the expression approaches Pi.

(Important Note: 180 is in DEGREES, not RADIAN)

Try N=1 million.
And you will see that it is indeed very close to Pi.

Observation circumstances
By observation of Maclaurin Series.

I also found another more tedious method to derive this, and it is based on assuming that a N-sided polygon is approximately a circle when N is large.

Note: I am actually in Singapore, but it is not in the list.

Information about this Observation
Peer-review ratings as of 06:21:42 on 23rd Oct 2017 (from 3 reviews, where a score of 100 is average):
Originality = 50.00, importance = 18.75, overall quality = 18.75

Published on Wednesday 26th July, 2006 at 12:14:04.

Creative Commons License
This work is licensed under a Creative Commons Attribution 2.5 License.
The full citation for this Observation is:
Doglas, Y. (2006). A New Approximation for Pi (3.141592654). PHILICA.COM Observation number 17.

Peer review added 30th August, 2006 at 23:34:31

Once again - incomplete sentence fragments, poor quality presentation, awful layout.

Peer review added 2nd October, 2006 at 16:34:53

Apart from issues on presentation quality, this Observation is fairly trivial. In fact, it can be extended to arbitrary numbers a ~ N sin(a/N).

Peer review added 3rd October, 2006 at 03:42:31

Sorry, this isn’t very impressive. I hope that most of my Calculus 2 students could do this. So is it a popularization? I don’t know; it’s not very clear. Keep trying, possibly in an area of Mathematics that hasn’t been so thoroughly worked?

Author comment added 3rd October, 2006 at 03:48:48

Actually, the beauty of this approximation is that an irrational number is obtained, from a trigonometrical function, and rational numbers.

Of course, one may notice that 180 degrees is actually Pi radians, so it may seem trivial. But I would like to think that this is a legitimate approximation to Pi, that is much shorter than other approximations.

Additional peer comment added 3rd October, 2006 at 05:03:34

It is actually erroneous to call this an approximation to pi at all, because you need the exact value of pi to do the conversion of 180 degrees to radians.




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