The area of a golden rectangle is 5 square feet. What is combined length of all 4 sides?
A golden rectangle has the property that the side lengths are in the ratio of 1 : [ 1 + √5]/2 = 1 : Phi
Let one side of the rectangle = S
Then we can let the other side = [1 + √5 ] / 2 * S = Phi* S
So we have that the area can be expressed as
S * [ 1 + √5] / 2 * S = 5
S^2 [ 1 + √5] = 10
S^2 = 10 / [ 1 + √5]
S = √ [ 10 / [ 1 + √5 ] ] =
So.....the combined length of all sides (the perimeter ) =
2 [ S + Phi * S] =
2 [ S(1 + Phi) ] =
2S [ 1 + Phi ] = [Note....1 + Phi = Phi^2 ]
2S [ Phi^2 ] =
2√ [ 10 / [ 1 + √5 ] ] [ 1 + √5]^2 / 4 =
( √10 / 2) [ 1 + √5]^(3/2) ≈
9.204 ft