# The Golden Rectangle and Fibonacci

One of the best things about teaching math is sharing all of the connections that exist between seemingly unrelated topics. Take, for example, the Golden Rectangle. Often described as the most visually “pleasing” quadrilateral, it is a rectangle whose length, l, and whose width, w, are in the the proportion $\displaystyle \frac{l}{w}=\frac{l+w}{l}$ .

Here’s an example:

4.69 / 2.9 = 1.62, and (4.69 + 2.9) / 4.69 = 1.62. The number they equal, 1.62, is called the Golden Ratio, and it has its own Greek letter, phi. It’s actually irrational, like pi and e, and it equals $\displaystyle \frac{1+\sqrt{5}}{2}$ . You can derive phi algebraically by solving $\displaystyle \frac{l}{w}=\frac{l+w}{l}$ for l in terms of w. From the quadratic formula, $\displaystyle l=\left( \frac{1+\sqrt{5}}{2} \right)w$, so $\displaystyle \frac{l}{w}=\frac{1+\sqrt{5}}{2}$.

A Golden Rectangle can be constructed with a straightedge and compass (or GeoGebra!):

1. Construct a square ABCD:

2. Construct the midpoint, E, of side AB.

3. Construct a circle with center at E, passing through C:

4. Extend rays through AB and DC. Construct the intersection point, F, of the circle and ray AB.

5. Construct a line perpendicular to ray AB, through point F. Construct the intersection point, G, of this perpendicular line and ray DC:

6. Quadrilateral AFGD is a Golden Rectangle.

Another way to construct a Golden Rectangle is to begin with two adjacent squares:

Add another adjacent square with dimension equal to the combined sides of the first two: