integration - Approximating a circle vs a diagonal.

Situation 1: A regular $n$-gon is inscribed in a circle. As $n$ increases without bound, the area of the $n$-gon approaches the area of the circle and the perimeter of the $n$-gon approaches the circumference of the circle.

Situation 2:
Consider a $1$ by $1$ square with one side labeled South and the other labeled North East and West as in a map.

A path is constructed from the Southwest corner to the Northeast corner.

If the path runs east on the south side for a distance $\frac{1}{2^n}$, then goes north for the same distance, then east again for distance $\frac{1}{2^n}$. And so on. Then the total length of the path is $2$

As $n$ increases without bound the area under the path and above the south side of the square approaches the area under the diagonal, but the length of the path remains $2$ and does not approach the length of the diagonal.

Why is there a difference?