My trig is pretty sketchy, pun intended, but here goes.

Using the law of cosines and assuming a unit circle with 30-gon side s, s^2 = 2-2cos(x) where x is angle formed by the two radii/legs.

Each of the 30 distance squares can be paired with one 180 degrees away. Since cos(x) = -cos(x+180), all the cosine sums cancel and we're left with 30*2 = 60 as the desired answer.