Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1
Let's note that P1P2+P2P3+P3P4+⋯+P9P10+P10P1 is simply the perimeter of the decagon
We also know that radius/2(−1+√5) is one of the sidelenghts. Thus, we can find the perimeter easily.
The perimeter is just
10(1/2)(−1+√5)=5(−1+√5)≈6.18
Thanks! :)