What is phi used for more than the golden ratio? What is the golden ratio?

The golden ratio is the positive root of x²=x+1

That is (1+sqrt(5))/2

https://en.wikipedia.org/wiki/Golden_ratio

In a certain sense, it is the most irrational number, meaning that its continued-fraction approximations are as bad as can be.

The letter phi is used for many things. It would be futile to try to give a complete list.

What is phi used for in calculus?

26 minutes ago, Brainee said:

What is phi used for more than the golden ratio?

Phi is most used for a general or unknown angle, both  in calculus and other uses of angles in Mathematics.

The Golden Ratio is also known as the Golden Section and appears in mny places in maths and natural science, as Joigus says.

 

Associated with the Golden section or ratio is the Golden Angle.

I like that phi is the limit of ratios of successive Fibonacci numbers.  What's cool is that the series will converge on phi with any starting value.