+116 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
+1804 Let's note that \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) is simply the perimeter of the decagon
We also know that \(radius/2 ( -1 + \sqrt {5})\) is one of the sidelenghts. Thus, we can find the perimeter easily.
The perimeter is just
\( 10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)
Thanks! :)
