1. A rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius.

The rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius.
1. To cover the rhombus, it is necessary to cover its four vertices. By the pigeonhole principle, at least one circle must cover two points, so $r \geqslant \frac{1}{2}$. On the other hand, it is possible to construct three circles with $r=\frac{1}{2}$ to cover the rhombus. Therefore, the smallest radius $r=\frac{1}{2}$.