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.

Angles in the SI: a detailed proposal for solving the problem

“…Various advantages of giving plane angles independent, dimensional status, rather than the dimensionless status that was formally adopted within the SI in 1995, have been set out many times over a long period [e.g. [8,9,[11][12][13][14]]. Such advantages are implicitly already recognised within the SI for quantities such as radiant intensity and radiance.…”

“…Since this paper was written, Brian Leonard's long, thoughtful and well-researched paper on the topic, Proposal for the dimensionally consistent treatment of angle and solid angle by the international system of units [14], has been accepted for publication. Although there is a great deal of common ground, it will probably be helpful to readers to highlight the main differences between his proposal and the one put forward here.…”

“…Of course, θ C = 1 rad, and it is technically correct that the unit symbol rad represents a quantity of angle. However, like many of the people referenced in [14], I find it very helpful to reserve unit symbols for the specification of quantities, such as 1.2 rad, and not to use unit symbols within equations. The symbol θ C , apart from looking like a term normally seen in equations, emphasises that the quantity is independent of the choice of angle unit.…”

Angular frequency in the International System of Units. Physicist’s view

“…Recommendation 1 of the CIPM 1980 was not accepted positively by all, see, for example, [5]. Recently, an increasing number of researchers come to the idea that the plane angle is a dimensional quantity, and the radian is not a dimensionless number one [6][7][8][9][10][11]. In [10], proceeding from the laws of physics and using rigorous mathematical calculations, it was shown that the plane angle does not depend on any physical quantity, has its own nonzero dimension, which, following [5,11], we will denote by the symbol A.…”

“…Recently, an increasing number of researchers come to the idea that the plane angle is a dimensional quantity, and the radian is not a dimensionless number one [6][7][8][9][10][11]. In [10], proceeding from the laws of physics and using rigorous mathematical calculations, it was shown that the plane angle does not depend on any physical quantity, has its own nonzero dimension, which, following [5,11], we will denote by the symbol A. This approach is fully consistent with the metrological rules for determining the dimensions of derived quantities and their units [3].…”