This is a two-part question, where I believe the first part is necessary (or at least useful) in solving the second part. (I'm also well aware that the answer may be far too complicated for the time I'm willing to invest in understanding it, but it will keep bugging me until I at least ask.)
First: consider a square unit grid $G_0$. The intersecting grid lines by definition form squares with area 1. Now take an identical grid $G_1$, apply a translation $(0\le a_1<1, 0\le b_1<1)$ and rotation $0\le \theta_1< \frac{\pi}{2}$, and overlay it on $G_0$. The intersecting lines now form polygons (example), and what I would like to know is the distribution $P_1(A)$ of their areas $A$ -- to be more precise, the answer to the question "pick a polygon at random; what is the probability of its area being $A$?".
If $\theta_1=0$, the problem has the same periodicity as the original grid and the answer is quite trivially $$P_1(A)=\frac{1}{4}\Big[\delta(A-a_1b_1)+\delta(A-a_1(1-b_1))+\delta(A-(1-a_1)b_1)+\delta(A-(1-a_1)(1-b_1))\Big].$$ For $\tan\theta_1\in\mathbb{Q}$, the periodicity increases and the polygons will no longer be rectangles, but the answer is still some finite sum of Dirac deltas and can be found by naïvely computing all polygons within one period. But for an arbitrary (almost always irrational) rotation, I am at a loss on how to even begin finding this distribution.
Now for the second part: instead of stacking just one grid atop the original, I want to stack $N$ of them, each with its own uniformly distributed random translation $(0\le a_n < 1, 0\le b_n<1)$ and rotation $0\le \theta_n<\frac{\pi}{2}$, and find the lowest integer $N$ such that, averaged over all realisations of the $a_n$, $b_n$ and $\theta_n$, $$\int_\alpha^1P_N(A) \mathrm{d}A<\varepsilon,$$ i.e. "how many randomly translated and rotated square unit grids must I stack, such that the probability of a polygon having an area greater than $\alpha$ drops below a threshold $\varepsilon$?"