The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?



Is $\pi=4?$
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 $2$.
If I were asked this question by a first-year calculus student, then the first thing I would do is write the following on the chalkboard:
$$ \sqrt{dx^2 + dy^2} \ne |dx| + |dy| $$
(This is still a common enough mistake among high school math students, but by the time anyone learns calculus, I hope that they would be aware that it is wrong.)

If you consider any infinitesimal element of the circle, the length is
$$d\ell = \sqrt{dx^2 + dy^2}$$
The only relationship between the circle and the "approximating" curve is that they have the same value of $|dx| + |dy|$, which is not a meaningful quantity when talking about length (at least, as has been pointed out, when you are using a Euclidean metric).  The value of $d\ell$ is not equal for any corresponding infinitesimal elements of the two curves, so there is no reason to believe it will be the same for the whole polygon.