The hexagon has area $6(\sqrt{3} / 4)(1)^{2}=3 \sqrt{3} / 2$. The region we want consists of six $120^{\circ}$ arcs of circles of radius $1 / 2$, which can be reassembled into two circles of radius $1 / 2$. So its area is $\pi / 2$, and the ratio of areas is $\pi \sqrt{3} / 9$.