Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)

Abstract Because of continued confusion caused by the SI’s interpretation of angle and solid angle as dimensionless quantities (and the radian and steradian as dimensionless derived units), it is time for the SI to treat these dimensional physical quantities correctly. Building on previous authors’ foundations, starting from Euclid’s Elements , I argue that angle should be recognized as a base quantity with an independent dimension: angle, A . A dimensionally consistent analysis of rotational geometry and mechanics results in the appearance of a constant of Nature equal to the central angle of a plane circular sector whose arc length is equal to that of its radius. This is the common (but not current SI) concept of the radian, rad, appropriately chosen as the base unit for angle. What the SI calls ‘angle’ is actually a nondimensionalized angle: the physical angle (dimension A ) divided by one radian. Using a coordinate-system-independent definition of solid angle, I show that this is a derived quantity with dimension A 2 and appropriate unit radian-squared. The steradian is retained as a coherent derived unit: sr = rad 2 . Clarification of terminology is needed in distinguishing between ‘geometrical’ and ‘mathematical’ trigonometric functions and in related topics, including the distinction between frequency and angular velocity, and between phase and phase angle, among other things.