Consider the Pick's theorem that states that the area $A$ of a polygon placed on a grid of equal-distanced points such that all the polygon's vertices are grid points, is given by
$A = i + \frac{b}{2} -1$,
where $i$ is the number of lattice points in the interior of the polygon and the number $b$ the number of lattice points on the boundary placed on the polygon's perimeter (see [1] for a picture).
Is there somewhere a similar relation that holds for a circle on a similar lattice ?