Number of points inside a shape on the Cartesian plane I was wondering if there is a formula which can help in finding number of points on a shape on the Cartesian plane, knowing that the shape isn't a rectangle nor a square.  
For example: consider an ellipse of equation $ax^2 + by^2 \le c$, Calculate number of points inside that ellipse (number of integer solutions for the equation).
 A: I know the following related facts. 
Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the numbers of grid points in the interior located in the polygon and on the boundary placed on the polygon's perimeter. 
A number of grid points inside a sufficiently fat figure appoximately equals its area. For instance, Gauss's circle problem asks for the number $N(r)$ of grid points inside the boundary of a circle of radius $r$ with centered at the origin. Gauss showed that $N(r)=\pi r^2+E(r)$, where $|E(r)|\le 2\sqrt{2}\pi r$,  see also this answer. Writing $|E(r)|\le Cr^\theta$, the best bounds on $\theta$ are $1/2<\theta\le 131/208\simeq 0.62981$. For the references  and the story of the bounds see Gauss's circle problem by Eric W. Weisstein at Wolfram MathWorld. There is also noted that  the problem has also been extended to conics, ellipsoids (Hardy 1915), and higher dimensions.
When I was a schoolboy at a contest I was given a problem to show that if an area of a figure is less than $1$ then there is a translated copy of the figure without grid points.
