TITLE: Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$
QUESTION [19 upvotes]: Possible Duplicate:
Is value of $\pi = 4$? 

Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?

REPLY [6 votes]: At 1:00 in the video, it is stated that $2(\frac{1}{2}\times\frac{\pi r}{2})=\pi r$, which is incorrect; the former expression simplifies to $\frac{\pi r}{2}$, not $\pi r$. The $\pi r$ line of reasoning is used to prove that $\pi=2$.
As for the proof that $\sqrt{2}=2$, bear this in mind.

(Great, now $2=4$. We're reinventing even more things than were anticipated now.)
It doesn't actually approach anything; it just looks like it does. Maybe it would help if the staircase were less crude; like, actually a reasonable approximation of the diagonal of the square, in which the median averages relatively near the diagonal.