With respect to \pi = 4, for the approximation to work you would need to use A = \pi r^2. i.e. A/r^2 = \pi. Initially, 1/0.25 = 4 but A is going to decrease with each tesselation (actually this appears to be more of a fractal) and eventually you will get to \pi.
You know some fractals take up a finite area but have infinite length.
Some volumes of revolutions have finite volume but infinite surface area? So you could fill it with paint but never paint it.