Cynthia Lanius

Golden Ratio Algebra

Golden Ratio 1.61803
Golden Ratio Table of Contents
  • Introduction
  • Find golden rectangles.
  • Build golden rectangles.
  • Confirm the ratio using algebra.
    • One property of golden rectangles is that their
      length    (length + width)
      
      ------ =  ----------------
      
      width         length       
      
      

    We used this property when we built the rectangle, and now we will use it to confirm the value of the golden ratio.

    When we cross multiply the above proportion, we get

    l2 = lw + w2 or

    l2 - lw - w2 = 0.

    Solving the equation gives us

    
    l = w(1 + sqrt5)
        ------------
            2   
    
    
    Divide both sides by w:
      l = (1 + sqrt5)
    ---  ---------
     w        2   
    
    
    
    
    Enter this into a calculator, and you'll see the approximation of the golden ratio -
    1.61803398874989484820

    There you have it, the derivation of the Golden Ratio.

    Golden Ratio Table of Contents
  • Introduction
  • Find golden rectangles.
  • Build golden rectangles.
  • Confirm the ratio using algebra.

  • Geometry Online Index
    Email any comments to lanius@math.rice.edu
    Copyright 1998-2008 Cynthia Lanius


    URL http://math.rice.edu/~lanius/Geom/algebra.html