The golden ratio satisfies:
a/b=b/(a+b) multiply both sides by (a+b)
(a^2+ab)/b=b multiply both sides by b
a^2+ab=b^2 subtract a^2+ab from both sides
b^2-ab-a^2=0 using the quadratic formula for expediency
b=(a±√(a^2+4a^2))/2 and we know b>0
b=(a+a√5)/2
b=(a/2)(1+√5)
If we let a=1
b=(1+√5)/2
So the golden ratio is (1+√5)/2
Since the longest side is 18in:
(1+√5)/2=18/s
s(1+√5)=36
s=36/(1+√5) in
s≈11.1 in (to nearest tenth of an inch)
