areas of parts of the circle in a grid

Question

Same colored cells have equal areas.
There is a grid consisting of NxN squares and a circle of Diameter N.
How do I find areas of the circle in each cell of the grid for any N . AFAIK, calculating 1/4 grid is enough because of symmetry.

Answer 1

Hint:

The real challenge is for incomplete cells, of course.

You need to solve the integral

$$\int_a^b(\sqrt{1-x^2}-c)dx,$$ where $a,b$ correspond to verticals on the grid or to the intersection of the circle and the horizontal $y=c$.

$$I=\frac12\left.(x\sqrt{1-x^2}+\arcsin x-2cx)\right|_a^b.$$

Answer 2

I will only work on the second given figure. The most difficult part is finding the area of the red region.

In the first supplied diagram above, $\angle POR = 60^0$, and A = [Orange region] = [sector OPR] – [⊿OSR].

By symmetry, B = A.

[Blue region] = $A \cup B$ = [sector OPQ] – [Green square]

[Red region] = $A \cap B = A + B - A \cup B$