Problem:
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
a+b2+c3+d6
where a,b,c, and d are positive integers. Find a+b+c+d.
Solution:
Position the 12-gon in the Cartesian plane with its center at the origin and one vertex at (12,0). Compute the sum, S, of the lengths of the eleven segments emanating from this vertex. The coordinates of the other vertices are given by (12coskx,12sinkx)
where x=30∘ and k=1,2,…,11. The length of the segment joining (12,0) to (12coskx,12sinkx) is
12(coskx−1)2+(sinkx)2=122−2coskx=24sin2kx
Thus the sum of the lengths of the 11 segments from (12,0) is
The same value, S, occurs if we add the lengths of all segments emanating from any other vertex of the 12-gon. Since each segment is counted at two vertices (its endpoints) the total length of all such segments is