Golden Rectangle

The Golden Rectangle

Back to the Middle Ages and the Renaissance

What is the Golden Rectangle?

The golden rectangle is a rectangle that can be cut into a square and a rectangle similar to the original rectangle. All around us, there are golden rectangles-in the shape of buildings, picture frames, doors, books, etc.

Leonardo Da Vinci and the Golden Rectangle

Leonardo Da Vinci, the Italian scientist, mathematician and artist, used golden rectangles to create his picture, Anunciation. He also wrote documents regarding the golden rectangle and the golden ratio, the ratio of the length of the golden rectangle to its width. Here is a proof of the Golden Ratio using the diagram on the left:

Step 1
ABCD ~ BCFE
Definition of Golden Rectangles

Step 2
AB/BC = BC/CF

Corresponding sides of
similar polygons are proportional

Step 3
x/1 = 1/(x-1)
Substitution

Step 4
x² - x = 1
Cross-Product Property

Step 5
x² - x - 1 = 0
Subtract 1 from each side

Step 6

Quadratic Formula
(if ax² + bx + c = 0) then:

Step 1 ABCD ~ BCFE	Definition of Golden Rectangles
Step 2 AB/BC = BC/CF	Corresponding sides of similar polygons are proportional
Step 3 x/1 = 1/(x-1)	Substitution
Step 4 x² - x = 1	Cross-Product Property
Step 5 x² - x - 1 = 0	Subtract 1 from each side
Step 6	Quadratic Formula (if ax² + bx + c = 0) then: