>>13868615A drawing doesn't prove anything.
Try to formalize the argument and find an analytic way to express the process of successively removing corners. You're probably going to run into some kind of contradiction at some point.
Also, the area of a circle is pi times the square of the radius (you can either prove this or you can DEFINE pi as being the area of the unit circle, and then you can use homotheties and properties of integrals to prove that in general the area of any given circle is equal to the area of the unit circle (that we define as pi) times the square of the radius of the given circle; I think Apostol defines pi in this way, if I recall correctly).
Given this fact, considering that the circle in your picture has diameter = 1, it follows that the radius = 1/2. So the are is:
Area = pi*1/4
If pi = 4, then Area = 1. But the square around the circle also has area = 1 since its sides are all equal to 1. Yet it's clear that the area of the square should be larger than the area of the circle since the circle is wholly contained in the square. So pi can't be equal to 4.
