The treatment of angles within the SI is anomalous compared with other quantities, and there is a case for removing this anomaly by declaring plane angle to be an additional base quantity within the system. It is shown that this could bring several benefits in terms of treating angle on an equal basis with other metrics, removing potentially harmful ambiguities, and bringing SI units more in line with concepts in basic physics, but at the expense of significant upheaval to familiar equations within mathematics and physics. This paper sets out the most important of these changes so that an alternative unit system containing angle as a base quantity can be seen in the round, irrespective of whether it is ever widely adopted. The alternative formulas and units can be treated as the underlying, more general equations of mathematical physics, independent of the units used for angle, which are conventionally simplified by implicitly assuming that the unit used for angle is the radian.

The radian and steradian are unusual units within the SI, originally belonging to their own category of ‘supplementary units’, with this status being changed to dimensionless ‘derived units’ in 1995. Recent papers have suggested that angles could be handled in two different ways within the SI, both differing from the present system. The purpose of this paper is to provide a framework for putting such suggestions into context, outlining the range of options that is available, together with the advantages and disadvantages of these options. Although less rigorously logical than some alternatives, the present SI approach is generally supported, but with some changes to the SI brochure to make the position clearer, in particular with regard to the designation of the radian and steradian as derived units.