Let the arc lengths be called $L_1$ and $L_2$ and the radii be called $r_1$ and $r_2$ where $L_1\le L_2$ and $r_1\le r_2$. Additionally let $d=r_2-r_1$ be the radial distance between the two arcs.
We have that the ratio of arc length to the radius is the angle, so
$$\theta=\frac{L_1}{r_1}=\frac{L_2}{r_2}$$ Hence we have $L_1r_2=L_2r_1$. Since we also know $r_2=d+r_1$, we have $$r_1L_2=L_1r_2=L_1d+L_1r_1\hspace{10mm}\implies\hspace{10mm}r_1=\frac{L_1d}{L_2-L_1}$$
Finally by applying $L_1r_2=L_2r_1$, we have $$r_2=\frac{L_2d}{L_2-L_1}$$