For radians, we can note that the length of a circle segment is rα, where r is the radius and α is the angle, where both the radius and circle segment have dime

For radians, we can note that the length of a circle segment is rα, where r is the radius and α is the angle, where both the radius and circle segment have dimensions length; then we must infer that α is dimensionless. Degrees follow a similar argument.