Three pentagram-shaped stars (the stars formed from the diagonals of a regular pentagon) are stacked up so that the bottom two ends of the tips touch the middle ends of the tips of the star below.
(See diagram.)

The distance from the top of the stack to the floor (where the bottom star's "feet" rest) is 4 feet.

What is the distance between the bottom two ends of the tips of the stack that touch the floor?

While I was cranking the last equation of the solution out on the calculator, I noticed an equivalence. I am looking for insight on this. It could be posed as a problem as,

Solve for x,

(cosx)^3 + (cosx)^2 = 1/8.

Of course I gave it away. The answer is 72 degrees (or 2*pi/5). But Im not sure why - my trig identities are lacking.