Please explain to me in simple terms what these circle intersections are all about.

Question

So, say you got 4 circles intersecting this way:

Now, I am looking for two things:

1. A proof that each part of the circle which is in an intersection is 1/4 the size of the whole circle's circumference
2. The exact area of the non-shaded region.

Now, in my search to finding the answer to this, I stumbled upon this article. The only problem? I have no idea what this article is trying to say, and how it can help me. I did find the equation to get the area of the shaded region ( it's A=2(pi-2) ) which I can use in Part 2 but I still don't understand how the solution got to there, and how to do Part 1. Please help me in learning what is trying to be said here in simpler terms! Thanks!

Answer 1

By your work an area of the non-shaded region it's $$4\pi-2\cdot2(\pi-2)=8.$$