Gauss Circle, Square And Triangle

 Consider the circle,

and the estimate for the number of integer points in the circle is, square minus four triangles. 

$e_z=(2r)^2-4(r(\sqrt{2}-1))^2$

$e_z=4r^2(1-(\sqrt{2}-1)^2)$

$e_z=4r^2(1-(2+1-2\sqrt{2})$

$e_z=8r^2(\sqrt{2}-1)$

$\pi$ need not get involved, but the estimation comes from a circle in a square.

Good night.