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
Suppose O is the centre of the circle. Then ΔOP1P2 is an isosceles triangle with OP1=OP2=1 and ∠P1OP2=360∘10=36∘.
Then P1P22=12+12−2(1)(1)cos36∘=2−1+√52=3−√52. Note that (√5−12)2=3−√52. Then P1P2=√5−12.
Repeating the same process 10 times gives P1P2+P2P3+P3P4+⋯+P9P10+P10P1=10⋅√5−12=5(√5−1).