Geometry! Help!

In order to solve this problem, we have to note something very important!

$P_1 P_2 P_3 \dotsb P_{10}$ forms a regular decagon!

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

In order to find one side length of the decagon, we can use the equation 

$\frac{\text{radius}}{2( -1 + \sqrt {5})}$

Using information from the problem, we can easily find the perimeter.

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

So 6.18 is about our answer!

Thanks! :)