September 25th, 2010, 06:27 PM  #1 
Newbie Joined: Jul 2010 Posts: 8 Thanks: 0  Pi Approximation
Hi All, I noticed an interesting trend that I thought I'd share... sin(360/2u) x u gets closer to pi as u becomes larger and larger; it is correct to about 8 places when u is 10^10. Any ideas? Marcel 
September 25th, 2010, 06:43 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Pi Approximation
I believe you have found the formula for the area of a regular polygon having 2u sides and circumradius 1. As u increases, the polygon more closely approximates a circle of radius 1, whose area is ?.

September 25th, 2010, 10:35 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Pi Approximation for x small. If you express your formula in radians, you'll see why it works...

September 25th, 2010, 10:45 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Pi Approximation
Very cool! I have seen that approximation used in physics for the linearization of dynamical systems, but that makes it very clear and easy to see why the formula converges to ?. 
September 26th, 2010, 08:35 AM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Pi Approximation Quote:
[color=#0040BF]3.1415926535897932384[/color]109662554790031839... where pi is 3.1415926535897932384626433832795028842... so it does have 20 correct decimals.  
September 27th, 2010, 11:49 PM  #6 
Member Joined: Sep 2010 Posts: 63 Thanks: 0  Re: Pi Approximation
It's an interesting result but of course using pi to compute pi is kind of circular.


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