Finding ratios of a pentagon ( diagonal to side)

Question

Given a regular pentagon, find the ratio of the diagonal, d, to its side, a.

Answer 1

interior angles sum = 180(n-2) = 180(5-2) = 540 degrees

each angle = 540/5 = 108 degrees

use half the diagonal as the base of a right triangle. top angle is 108/2 = 54 degrees

bottom angle = 36 degrees

ratio of base to the hypotenuse = cos36

base = half the diagonal, hypotenuse = pentagon side = a

d/a = 2cos36 = about 2(.81) = 1.62

d/a = 162/100 = 81/50