$\left[\begin{array}{l}{[\text { Tangent Circles }} \\ {[\text { Similar Figures }]}\end{array}\right]$

Several circles are inscribed in an angle, their radii increasing. Each subsequent circle is tangent to the previous one. Find the sum of the lengths of the second and third circles, if the radius of the first is 1, and the area of the circle bounded by the fourth circle is $64 \pi$.
Since the area of the circle bounded by the fourth circle is $64 \pi$, the radius of the fourth circle is 8. Let $r$ and $R$ be the radii of the second and third circles, respectively. Note that the figures consisting of two adjacent circles are pairwise similar. Therefore, $\frac{\tau}{1}=\frac{R}{\tau}=\frac{8}{R}$, from which we find that $r=2, R=4$.

Thus, the sum of the lengths of the second and third circles is $2 \pi(r+R)=2 \pi \cdot 6=12 \pi$.

## Answer

$12 \pi$.