Geometry

Let's note something really important first. 

$P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1$ is simply the perimeter of a regular decagon. 

We can find one side of the decagon using the equation $radius/2 ( -1 + \sqrt{5})$

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

So our answer is about 6.18. 

Thanks! :)

~NTS