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 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 $\sin x\approx x$ 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
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Re: Pi Approximation

Quote:
 Originally Posted by MarkFL 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 ?.
If you take the next asymptotic term, sin x = x + x^3/6 + O(x^5) for x near 0. This can be used to estimate the error in the approximation. So when x is 10^10, the error is about (pi/10^10)^3/6 * 10^10 = pi^3/6 / 10^20 which is about 10^-20, so we expect the approximation at 10^10 to have about 20 decimal places. Indded, it is
[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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