The staircase paradox, or why $\pi\ne4$

This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is $2$, then take shorter and shorter stair-steps and the length is $2$ but your path approaches the diagonal. So $\sqrt{2}=2$.

In both cases, you are approaching the area but not the path length. You can make this more rigorous by breaking into increments and following the proof of the Riemann sum. The difference in area between the two curves goes nicely to zero, but the difference in arc length stays constant.

Edit: making the square more explicit. Imagine dividing the diagonal into $n$ segments and a stairstep approximation. Each triangle is $(\frac{1}{n},\frac{1}{n},\frac{\sqrt{2}}{n})$. So the area between the stairsteps and the diagonal is $n \frac{1}{2n^2}$ which converges to $0$. The path length is $n \frac{2}{n}$, which converges even more nicely to $2$.

R.I.P. Archimedes

A photogenic answer to such a question!

This problem illustrates the fact that two functions can be very close: $|f(x)-g(x)|<\epsilon$ for all $x\in [0,1]$, but their derivatives can still be far apart, $|f'(x)-g'(x)|>c$ for some constant $c>0$. In our case, let $x=a(t),y=b(t),0\le t\le 1$ and $x=c(t),y=d(t), 0\le t\le 1$ be the parametrizations of the two curves. By smoothing the corners, we may assume that both are smooth. $$ \|(a(t),b(t))\|\approx \|(c(t),d(t))\|$$ does not imply $$ \|(a'(t),b'(t))\|\approx \|(c'(t),d'(t))\|$$ Therefore $\int_0^1 \|(a'(t),b'(t))\| dt$ need not be close to $\int_0^1 \|(c'(t),d'(t))\| dt.$ Here $\|(x,y)\|$ denotes $\sqrt{x^2+y^2}$.