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avatar+1121 

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

 Mar 14, 2024
 #1
avatar+128794 
+1

This is  just the perimeter of  a  decagon

 

One side, s, can  be  found by the Law of Cosines

 

s^2 =  1^2 + 1^2   2(1*1)cos 36°

 

s^2  = 2  -2 cos (36°)

 

s = sqrt  [ 2 - 2 cos (36° ]

 

Perimeter  = 

 

10 sqrt [ 2  -2 cos (36°)]  ≈  6.18

 

cool cool cool

 Mar 14, 2024

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