Okay, the reason why it’s wrong? It’s in the video, but if you ignore videos, here is the answer.
Zoom in on one of the corners of the square with a circle part in it.

So the ‘circle’ is not even a circle with perimeter pi, but is actually an ‘infinigon’ (cleverly stated by Vi Hart – oh wait, it’s in Wiktionary, so probably not) with perimeter 4.
๐๐๐๐๐คฃ๐๐๐๐๐๐;-);-):-)^_^ยฏ\_(ใ)_/ยฏ๐๐๐๐๐๐๐๐๐๐คฃโ๐๐๐๐