# How to cover a disk with radius $1.01$ with three unit disks?

I need to show that one can cover the disk of radius $$1.01$$ with three distinct unit disks. But after trying around a bit I am not sure that this is even possible. Could you please give me a hint how to do this?

• Where is this exercise from? I don't see how this is possible as well. Oct 14, 2020 at 9:44
• They can't be disjoint. They can be distinct, though. Oct 14, 2020 at 9:47
• Ok, I just got an email, it was a typo, the disks are supposed to be distinct, not disjoint. Thanks for your efforts. Oct 14, 2020 at 9:49

If you offset the unit disk centers by $$d$$ in symmetric directions, the circles intersect in pairs at a distance $$r$$ of the origin such that

$$\left(r-\frac d2\right)^2+\frac{3d^2}4=1.$$

The relevant root is

$$r=\frac{\sqrt{4-3d^2}+d}2$$ and it achieves a maximum when $$d=\frac1{\sqrt3},$$ corresponding to

$$r=\frac2{\sqrt3}>1.01$$

The minimum decentering is obtained with

$$\left(1.01-\frac d2\right)^2+\frac{3d^2}4=1,$$

or $$d\approx0.02031$$