If the Golden Ratio phi is irrational, why is it called the Golden Ratio?

The Golden Ratio is basically a ratio between two given numbers. If this ratio happens to be the irrational number 1.6180339887 etc, the ratio is called the Golden Ratio.

Because the definition is any pair of numbers that satisfiess the ratio a/b = b/(a+b).

Solving it (taking a=1) does get an irrational result though. (1+sqrt[5])/2.

One of the first places the golden ratio appeared was as part of the Golden Rectangle (a rectangle that can be divided into a square plus a smaller copy of the original rectangle -- see the picture at right in the link).

Golden rectangles can come in any size: if you double all the sides of a golden rectangle, you'll be left with a (bigger) golden rectangle. What's important is the ratio between length and width. And that ratio is the golden ratio.