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.