show that the area of a regular n-gon(polygon) is the same as that of a circle as $n\to \infty$

Question

ganeshie8 · Answer

to prove :

$\large \lim \limits_{n \to \infty}~\dfrac{1}{2} n r^2 \sin(\frac{2\pi}{n}) = \pi r^2$

anonymous · Answer

yes :)
 seems a bit obvious from here

ganeshie8 · Answer

yes !   sinx/x limit leads to pir^2 i hope should be easy...  :)

anonymous · Answer

ohh yeah,very easy,and the next question goes>>> find the ratio of A:n if each side has length $p$ and distance from center to corner is $r$

ganeshie8 · Answer

we want to find below 

 Area
------
  n


is it ?

anonymous · Answer

yes ,yes,but we no longer focus on the situation n to infty

ganeshie8 · Answer

just divide n, we're done right ?

ganeshie8 · Answer

A/n =   $\large \dfrac{1}{2}  r^2 \sin(\frac{2\pi}{n}) $

?

anonymous · Answer

yes $$\frac{A}{n}=\frac{1}{2}\sin \frac {2\pi}{n}r^2$$ is not enough since we still have n as a variable inside sin

anonymous · Answer

we want it expressed in terms of $$r,p$$

ganeshie8 · Answer

ohkk got u

ganeshie8 · Answer

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