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.
