A golden rectangle has the property that it produces a smaller rectangle of the same proportions when a square of the shorter side is removed. If a and b are the short and long sides of a golden rectangle then (b−a)/a = a/b, that is b/a − a/b = 1. If we multiply this equation by b/a we get (b/a)² − 1 = b/a, whence b/a = (√5 + 1)/2 = 1.6180339... = F, while multiplying by a/b we get 1 − (a/b)² = a/b, whence a/b = (√5 –1)/2 = 0.6180339... = f, where F and f are the golden ratios.
These two ratios are related by: F.f = 1 = F − f.
We also have F² = 1 + F = 2 + f and f² = 1 − f = 2 − F.
There are two golden triangles both isosceles: the 72°-72°-36° triangle with side/base = F, and the 36°-36°-108° with base/sides = F. Since 108° is the vertex angle for a regular pentagon, and 36° is the vertex angle for a regular pentagram, the golden ratio is evident in these figures too.