# Exploring the golden ratio

The golden ratio was mentioned in Euclid’s Elements around 300 BCE, although it was probably known even earlier than that. It has been studied by mathematicians ever since.

At its simplest, we can say that two quantities are related by the golden ratio if the ratio between them is the same as the ratio between their sum and the largest of them. We can express this as a simple formula. Two numbers *a* and *b* (where *a > b*) are in the golden ratio if:

We can also show this as a diagram:

The first rectangle is size *a* by *b*, and the second is *a + b* by *a*. If *a/b* is the golden ratio then the two rectangles will be similar (in the mathematical sense of being the same shape but not necessarily the same size). We call this rectangle a *golden rectangle*.

Notice that the larger rectangle is formed by adding a square to the smaller rectangle. The two rectangles will only be similar if the original rectangle has sides in the golden ratio.

# What is the value of the golden ratio?

Looking at the smaller rectangle, the value of the golden ratio is given by the long side divided by the short side (by convention, we do it that way round to get a value greater than one). We will call that ratio ϕ (the Greek letter *phi*):