One way to calculate how many there would be is to add 1 for every point on the grid where the distance to the center is greater than the radius, and count how many times that happens checking every grid point. But that's really more of an algorithm...
I found this other way:
For example, this sum totals 113 for r=6 which is the correct value.
The final expression is totalling these values up and solving for how many must be inside the whole circle from that sum.
Unfortunately it's harder to get rid of the absolute value and it didn't happen that there was a closed form sum for the double sum over the grid, but I still think it's an interesting approach.