Cut a disk into $N$ pieces to best pack into a square I have a 3d printer with a square boundary. I'd like to print something with a circular base. It occured to me that I could print a circle with a diameter bigger than the width of my square if I broke it into semicircles.
If you are happy to print off on piece at a time, you can fit a semicircle whose diameter is $1/2 + 1/\sqrt2$ into the square.
What if you wanted to print off both bits at the same time? What's the biggest pair of semicircles your could print on a square plate? How would you arrange them? Is a pair of semicircles the best shape? 
What if you were ok with slicing your shape into three or even four pieces?
At the end of the day - I'll just ask a mate of mine with a bigger printer to print my object. But, it's possibly an interesting dissection with some useful applications.
 A: Now that's a bunch of interesting questions.


*

*We have a unit square and a circle of diameter 1 (Fig. 1), and we aren't quite happy with that.


*If we are doing it in two parts, we may print a half of a bigger circle (Fig. 2). IMHO, this is $4-2\sqrt2\approx1.172$; I fail to see how would you achieve ${1\over2}+{1\over\sqrt2}\approx1.207$, but this is pretty big anyway. In fact, it is so big that a whole circle of this diameter would have an area greater than 1, so we may just as well forget about printing it in one turn.

*Now what if we want to squeeze all parts in one square? The upper bound for diameter is $\sqrt{4\over\pi}\approx1.128$. Again, I fail to see how would you fit two semicircles (or any other two parts, for that matter) of a circle with diameter even slightly greater than 1. As for 3 pieces, here's the best I could find (Fig. 3). This is $\sqrt6-\sqrt2\approx1.035$, which is not much of an achievement, but still somewhat greater than 1. Let those who can do better.
