Golden Ratio

The Golden Ratio is the ratio of two adjacent sides of a rectangle that the ancient Greeks held to be the most aesthetically pleasing when applied to building. The Classical Greek Parthenon, near Athens, embodies the golden ratio in much of its construction.

One definition of the golden ratio is that "A rectangle conforming to the golden ratio when cut to leave leave a square will also leave another rectangle conforming to the golden ratio". Such a rectangle is often called a "golden rectangle".

The image below shows a rectangle conforming to the golden ratio. The rectangle conforming to the golden ration is shown in the colour purple. When a square is cut from this golden rectangle, it leaves a rectangle shown in the colour blue. By moving the blue cut rectangle and then drawing a diagonal line from the corner to corner of the original golden rectangle (shown in purple), it is clear to see that the cut rectangle (shown in blue) has the same ratio as the original rectangle.

Calculating the Golden Ratio

Calculating the Golden Ratio is very easy. Using the diagram on the right, the golden ratio (φ) is defined as a/b as this is a golden rectangle. golden rectangle Now, we know that if a square is cut from the golden rectangle, a golden rectangle remains. So, if c=b, which is the square to be cut off, the dimensions of the remaining rectangle is (a-b) by b. As we know this to be a golden rectangle, then b/(a-b) = φ which equals a/b. Putting this in mathematical notation:

φ = b/(a-b) = a/b

or by rearranging

1 = a(a-b)/b2

and by expanding

1 = (a2-ab)/b2

on simplifying

1 = (a/b)2 - a/b

or by rearranging

0 = (a/b)2 - a/b -1

and as φ = a/b,

0 = φ2 - φ -1

which is a familiar quadratic equation the solution which is:

φ = 1 ± √(1+4)


which gives φ = 1.618034




Terms of Use of this Web-site

The following uses of this service are prohibited:

  • Use by a non-human
  • Any illegal activities
  • Using it to process pornographic images
  • Using it to process images which may be illegal in any country of the world

We may pass details of usage of this site to the police.

We reserve the right to remove any link without notice.

This service is provided without warranty of any kind.

This service is provided on an as-is basis.

This service may be withdrawn temporarily or permanently without notice.

tau 10

Valid XHTML 1.0! Valid CSS!