Geometry

Let's first notice something really important. 

The list $P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1$ are just the sides of a regular decagon!

The sidelength of one side the decagon can be represented as $\frac{\text{radius}}{2 ( -1 + \sqrt{5})}$

Plugging in all the values we know, we have

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

So about 6.18 is our answer!

Thanks! :)