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# golden rectangle

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The area of a golden rectangle is 5 square feet. What is combined length of all 4 sides?

Aug 26, 2019

#1
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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   Aug 27, 2019
#3
+1

Let's make sure that that's correct:

9.204 / 2 = 4.602

Ratio is  1 : 1.618        ratio in percentage>   38.2%  :  61.8%

SS--short side     LS--long side

4.602 / 100% = SS / 38.2%                  4.602 / 100% = LS / 61.8%

SS = 1.754964                                      LS = 2.844036

1.754964 * 2.844036 = 4.991180                   c o r r e c t ! ! !

Guest Aug 28, 2019
#4
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Shorcut >>>

9.204 / 2 = 4.602

4.602 * 0.618 = 2.844036

2.844036 * 0.618 = 1.757614

2.844036 * 1.757614 = 4.998718 ✔ Guest Aug 28, 2019
edited by Guest  Aug 29, 2019