Three concentric circles are shown. The two largest circles have radii of $12$ and $13.$ If the area of the ring between the two largest circles equals the area of the smallest circle, determine the radius of the smallest circle.

[asy]
import graph;
filldraw(circle((0,0),12), lightgray, black+linewidth(1));
filldraw(circle((0,0),10), white, black+linewidth(1));
filldraw(circle((0,0),6), gray, black+linewidth(1));
dot((0,0));
[/asy]
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 $5.$