Math puzzles!

This has gotten me very intrigued. How does that work out? I'm a little sad that I can't figure out what's wrong with the logic.

You can calculate the difference in the perimeters at each step... that sequence doesn't converge to zero. It's just the sequence: 4-pi, 4-pi, 4-pi, 4-pi..... so the error is always 4-pi even after you "repeat to infinity". Likewise the process from the 4th to the 5th panel is technically impossible. It'll never be smooth enough using that method. So you can never say that they are equal, even in the limit. It's not like the n-side polygon inside of a circle trick.... where you let n go to infinity and you get the circle. That one DOES smooth out enough.... but it's really hard to explain without getting into slightly complicated math.