jmc

The area of the ring between the two largest circles is $$\pi \cdot 13^2-\pi \cdot 12^2=169\pi -144\pi =25\pi .$$Suppose that the radius of the smallest circle is $r.$ Thus, the area of the smallest circle is $\pi r^2.$ Since the area of the smallest circle is equal to the area of the ring between the two largest circles, then $\pi r^2=25\pi$ so $r^2=25$ and so $r=5$ since $r>0.$

Therefore, the radius of the smallest circle is $\boxed{5}.$