## 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