Finding Circle Circumference from Inscribed N-Sided Polygen

[logan3]

The perimeter P of a regular polygon of n sides inscribed in a circle of radius r is given by $P = 2nr sin (180^o / n)$.

I was curious whether it's possible to approximate the circumference of a circle by taking the limit as n goes to infinity of the above perimeter equation is some way?

Thank-you

 

[BiGyElLoWhAt]

I would think it should be possible. As it stands, applying that limit would give you infinite*0.
Is there anyway you could express the same function in terms of other trig functions that would give you inf/inf or 0/0? Then you could potentially apply l'hopital's rule and get something useful.

 

[logan3]

$P = 2nr \sin (180^o / n) = \frac {2nr} {\csc (180^o / n)} = \frac {2nr} {\csc (180^o / n)}$
$\displaystyle \lim_{n\rightarrow \infty} {\frac {2nr} {\csc (180^o / n)}} = \frac {\infty} {\infty}$

L'Hôpital's rule:
$\displaystyle \lim_{n\rightarrow \infty} {\frac {f'(n)} {g'(n)}} = \displaystyle \lim_{n\rightarrow \infty} {\frac {2r} {-\csc (180^o / n) \cot (180^o / n)}}$

Sorry, this is about as far as I got right now. Also, I'm not sure where the $\pi$ is going to come in for the $C = 2\pi r$, though I assume it will have to come from the trig functions somehow.

 

[mathman]

$\lim_{n->\infty}\frac {sin(\frac{\pi}{n})}{\frac{\pi}{n}}=1$.
Therefore $\lim_{n->\infty}2nrsin(\frac{\pi}{n})=2\pi r$.

 

[logan3]

Where's the denominator of pi / n come from in the second expression?

 

[Svein]

See http://en.wikipedia.org/wiki/Pi.
Scroll down to "Polygon approximation era".

 

[micromass]

It works in this case. But you need to be careful with polygonal approximation of lengths, it might not always give the right answers:

 

[BiGyElLoWhAt]

It works in this case. But you need to be careful with polygonal approximation of lengths, it might not always give the right answers:

Hehe. :D

 

[mathman]

Where's the denominator of pi / n come from in the second expression?

P=2nrsin(π/n)=2πr(sin(π/n)/(π/n))

 

[logan3]

I don't understand, sorry.

But it seems like my thoughts about this relationship were in the right direction.

 

[BiGyElLoWhAt]

Its just algebra that he did to get it into a new form. Multiply by pi/pi, and kick the n into the denominator of the denominator. Now you have sin (x)/x.

 

[LCKurtz]

And, of course, $\lim_{x\to 0}\frac{\sin x}{x}=1$ only if $x$ is measured in radians.