Paul Quincey makes a compelling argument for recognizing angle as a base quantity with the radian as the base unit. Solid angle is then a derived quantity with the steradian a coherent derived unit equal to one square radian. The author demonstrates how familiar equations of rotational motion appear to result from dimensionally consistent explicit-radian equations by ‘setting the radian equal to one’—which he calls the radian convention. Quincey also claims, based (solely) on assumed analogies with translational motion, that for rotation, the so-called ‘improved’ units for torque, angular momentum and moment of inertia must be J/rad, J/(rad/s) and J/(rad/s)2, respectively, and that the conventional units (N m, kg m2 s−1 and kg m2) result from application of the radian convention to these quantities. However, based on fundamental physical principles, I show here that, although the radian convention may help in understanding a confusing notational change applied to angular displacement and its time derivatives when comparing explicit-radian equations to their equivalent familiar forms, it cannot be applied to torque, angular momentum or moment of inertia. The dimensionally correct SI units for these quantities are, respectively, the well-established angle-independent units: newton metre, kilogram metre-squared per second and kilogram metre-squared.