I think it's because no matter how many corners you cut it's still an approximation of the circumference area. There's just an infinite amount of corners that sticks out

It's a fractal problem, even if you repeat the cutting until infinite, there are still a roughness with little triangles which you must add to Pi, there are no difference between image 4 and 5, the triangles are still there, smaller but more. But it's a nice illusion.

Because you never make a circle. You just make a polygon with a perimeter of four and an infinite number of sides as the number of sides approaches infinity.

https://youtu.be/VYQVlVoWoPY