I think as the number of corners are removed and approach infinity, the area of the shape that was once the square *will* approach the exact area of the circle, however, the shape is not a circle, it's an infinite number of infinitesmal corners who's total length with always be 4.

If you were to travel along the shape, you'd never be travelling tangent to the circle (well except when the corners are tangent) as you would if this shape converged as a circle, but would instead be traversing an infinite number of corners, up and down up and down...