+1557 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
+1926 Let's first notice something really important.
The list \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) are just the sides of a regular decagon!
The sidelength of one side the decagon can be represented as \(\frac{\text{radius}}{2 ( -1 + \sqrt{5})}\)
Plugging in all the values we know, we have
\(10 (1/2) ( -1 + \sqrt{5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)
So about 6.18 is our answer!
Thanks! :)
+1926 Let's first notice something really important.
The list \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) are just the sides of a regular decagon!
The sidelength of one side the decagon can be represented as \(\frac{\text{radius}}{2 ( -1 + \sqrt{5})}\)
Plugging in all the values we know, we have
\(10 (1/2) ( -1 + \sqrt{5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)
So about 6.18 is our answer!
Thanks! :)
