Frustrating area calculation

First of all, hello, I'm new, and I make awful sketches on the computer. Now, for the problem.

http://i.imgur.com/6TX84.jpg

This is roughly the sketch I was given (they're supposed to be perfect quarter circles going from the edges of the square to the opposite one). The length of a side is 1. What I need to know is the middle part Z's area (and then find the percentage of it compared to the full square). I've been sitting for over a week trying to figure this out, but I'm going in circles and it's really starting to confuse me.

What I have so far (not that much):

3x + 2y + z = π/4

x + 2y = 1 - (π/4)

2x + z = (π/2) - 1

x - 2y + z = (3π/4) - 2

4y - z = π - 3

After this, I am incredibly confused. I've tried everything I know to solve this, but to no avail. I'm betting the answer really is super simple and I just can't find it...

Thanks on beforehand for help! :)

//Kraetyz