Dew Drops

In the right triangle $OEA$ we have $AE=1$ and $OA=2$ so $\angle AOE=30^\circ$ and as a result $\angle EAF=120^\circ$ .

In triangle $AED$ the sides are $AE=1, AD=1+R$ and the angle $a=arccos(\frac{1}{1+R})$ .

Triangles $ADG, AGC,$ and $ACF$ are all congruent. The angle $b=arcsin(\frac{R}{1+R})$ .

The angle $EAF=a+3b= arccos(\frac{1}{1+R})+3 arcsin(\frac{R}{1+R})=120.$

This is an equation in $R$ with the solution being $R\approx0.626$ .