Geometry

First, let's call the center of the circle, $C$. Essentially, all the sides are equal ($P_1 P_2, P_2 P_3, P_3 P_4$, etc.) since this is a regular polygon so we just need to find one of those sides.

 

Connect $C$ to $P_1$ and $P_2$, and those lengths are $1$ since the radius of the circle is $1$.

 

Now, we need to find the central angle between those two sides. Since this is a decagon, the central angle is $360/10$, which is $36^{\circ}$.

 

To find $P_1 P_2$, we can use Law of Cosines: $(P_1 P_2)^2=1+1-2*1*1*\cos(36^{\circ})$. When you solve for $P_1 P_2$, you get $P_1 P_2 \approx 0.618$.

 

Since all the sides are equal, the answer is just $10(P_1 P_2)$, so the answer is approximately $6.18$.