Sequence of loops that converges to the unite circle with length greater than 2\pi

Question

I would like to know how to construct a sequence of loops that converges to a circle in the Hausdorff distance, but has constant (or increasing) length greater than the circumference of the circle.

Answer 1

The curve with polar representation $$r(\phi)=1+{2\over n}\cos(n\phi)\qquad(0\leq\phi\leq 2\pi)$$ goes $n\gg1$ times back and forth between the circles $r=1+{2\over n}$ and $r=1-{2\over n}$, hence has length at least $$2n\cdot{4\over n}=8>2\pi\ .$$