Geometry

Let's note that $P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1$ is simply the perimeter of the decagon

We also know that $radius/2 ( -1 + \sqrt {5})$ is one of the sidelenghts. Thus, we can find the perimeter easily. 

The perimeter is just

$10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18$

Thanks! :)