December 12th, 2018, 01:26 PM  #1 
Member Joined: Feb 2018 From: Canada Posts: 46 Thanks: 2  Richardson extrapolation
Consider a regular nsided polygon tangential to and enclosing the unit circle to approximate $\pi$. a) Use geometrical arguments to show that the halfperimeter of the polygon $C_n=n\tan(\pi /n) \geq \pi$. b) Calculate $C_4$ and $C_8$. c) Use Richardson extrapolation with $h = 1/n^2$ to improve upon your results. For part b) I've got: \[ C_4 = 4\tan(\frac{\pi}{4}) = 4\times 1 = 4, \] \[ C_8 = 8\tan(\frac{\pi}{8}) = 8 \frac{\sin(\frac{\pi}{8})}{\cos(\frac{\pi}{8})} = 8\frac{\frac{\sqrt{2\sqrt{2}}}{2}}{\frac{\sqrt{2+\sqrt{2}}}{2}} = 8\frac{\sqrt{2\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\] By drawing, I have been able to do the part a) but I have no clue how to do part c). Can someone provide me some help? Thank you. Last edited by skipjack; December 12th, 2018 at 07:00 PM. 
December 12th, 2018, 01:29 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,355 Thanks: 737  
December 12th, 2018, 07:04 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,921 Thanks: 2203 
tan($\pi$/8) = √2  1


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extrapolation, richardson 
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