Geometry

First, let's note something really important for the problem. 

We have $P_1 P_2 P_3 \dotsb P_{10}$, which actually just forms a regular decagon!

$P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1$is just the perimeter of that decagon. 

The length of a decagon can be found with formula $\frac{\text{radius}}{2( -1 + \sqrt {5})}$

Thus, the perimeter is $10 (1/2) ( -1 + \sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18$

So our answer is approximately 6.18

Thanks! :)