Following problem: I want to approximate the number of grid points in a polygon, based on the condition that the distance of the grid points are variable. What i need is an approximation, i am aware that a precise information is hard to obtain.
I first started experimenting with several numbers to calculate the number of grid points in a square and developed a formula: $$ (\frac{\sqrt{x} }{y} +1 )^{2} $$ x is the surface area and y is the distance of each grid point from each other.
(quick question: is there an already existing formula for or a theorem?)
Anyway i want to approximate the number of grid points with a distance of 7 Kilometers based on the area of Germany, which is about 357.000 square kilometers. Therefore i thought of drawing the map of germany in a coordinate system with integer coordinates and use Pick's theorem, but my map must be huge if a want a distance of 7km for each point and the job would be very labourious.
Is there a smarter solution? Like mentioned before, i need a realistic estimate not a exact number. Thank you very much.