What you must do is look at whether a quantity is scale invariant or, more generally, invariant with respect to a change of units. Angles have this property: they are defined as a ratio of lengths that both scale in proportion to the geometric figure. A uniform dilation of a circle, sphere or any other geometric figure (equivalent to multiplying our length units by a conversion factor from meters to feet of feet to Standard Snozfurgles) leaves the ratio of any two distances between any two pairs of points unchanged.

The same is not true of the ratio of a dimensioned length and the unit length. Represent the length as a line segment on a co-ordinate chart. Dilate the co-ordinate chart as above and watch it shrink / grow. Now, the unit length does not scale in the same way: it is defined in terms of a physical length: a unit measuring rod, a number of wavelengths and so forth. These natural things do not change with arbitrary dilations we choose to make on our co-ordinate systems.

You must check whether a quantity is scale invariant or, more generally, invariant with respect to a change of units. Angles have this property: they are defined as a ratio of lengths that both scale in proportion to the geometric figure. A uniform dilation of a circle, sphere or any other geometric figure (equivalent to multiplying our length units by a conversion factor from meters to feet of feet to Standard Snozfurgles) leaves the ratio of any two distances between any two pairs of points unchanged.

The same is not true of the ratio of a dimensioned length and the unit length. Represent the length as a line segment on a co-ordinate chart. Dilate the co-ordinate chart as above and watch it shrink / grow. Now, the unit length does not scale in the same way: it is defined in terms of a physical length: a unit measuring rod, a number of wavelengths and so forth. These natural things do not change with arbitrary dilations we choose to make on our co-ordinate systems.