What is the area of the intersection of two rings? 
A ring is two concentric circles. There are two rings whose centers are a distance D apart. The outer and inner radius of the rings are $r_a$, $r_b$, $r_c$, $r_d$.
So $r_a > r_b$ and $r_c > r_d$. Let's also assume that the thicknesses of each ring are the same, so $r_a - r_b = r_c - r_d$.
Note that $r_a$ does not necessarily equal $r_c$! I just drew it that way in the picture for simplicity. There is also no need to make the outer radius of one ring touch the center of the other ring. D could be anything.
What is the area of one of the red "diamonds"? Do we have enough info to determine this? If so, I cannot figure it out. Obvious the area of each ring is $\pi (r_a^2 - r_b^2)$ and $\pi (r_c^2 - r_d^2)$, but I want the intersection of them, not the whole ring.

You can find a formula for the area of intersection between two circles here. Using only this formula, you can find the area you are looking for. 

In the figure above I've drawn the scenario you describe, with names for the various areas which make up the intersection area between the circles. If we denote the Area of Intersection between the two circles with radii $r_1$ and $r_2$ as $AI(r_1,r_2)$, we see that: $$A_2 + 2 \cdot A_1 = AI(r_c,r_a) - AI(r_d, r_a) = C_1$$
$$A_3 + 2 \cdot A_1 = AI(r_a,r_c) - AI(r_b, r_c) = C_2$$
$$A_2 + A_3 + 2 \cdot A_1 = AI(r_c,r_a) - AI(r_d, r_b) = C_3$$
Therefore, $$C_1 + C_2 - C_3 = 2 \cdot A_1$$
or $$A_1 = \frac{C_1 + C_2 - C_3}{2}$$
