A dual concept of the angle in mathematics and practice

We consider the angle in mathematics and arrive at a conclusion that there are two concepts on the issue. One is a descriptive geometrical one, while the other is from functional analysis. They are somewhat different, allow for different options, and both are legitimate and in use. Their difference may cause certain confusions. While the geometrical angle' allows for different choice of units, the functional angle’ is a purely dimensionless one, being related to the angle in radians. We consider possible options to resolve the problem as it concerns the units.