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September 25th, 2010, 06:27 PM   #1
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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
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September 25th, 2010, 06:43 PM   #2
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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 ?.
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September 25th, 2010, 10:35 PM   #3
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Re: Pi Approximation

for x small. If you express your formula in radians, you'll see why it works...
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September 25th, 2010, 10:45 PM   #4
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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 ?.
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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.
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September 27th, 2010, 11:49 PM   #6
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Re: Pi Approximation

It's an interesting result but of course using pi to compute pi is kind of circular.
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