# 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.

• See this question. Commented Dec 26, 2020 at 13:07

2. 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.
3. 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.