Angles—Let’s treat them squarely

Abstract

We suggest a self-consistent treatment of the dimensions and units of the geometric quantity “angle.” The method regards “angle” as a fundamental dimensional physical quantity, on a par with length, mass, time, etc. All units (whether angular or otherwise) are treated on an equal footing and balance out correctly; in particular, “radian” units need never be spuriously inserted or deleted. The method could find application in algebraic and calculus symbolic manipulation computer programs to correctly process units of physical quantities. The technique necessitates a minor modification of the relation “s=Rθ” and its consequences, rather than any modification of the units of other physical quantities (such as moment arms) as previously suggested by others. We make several important clarifying distinctions: (a) ω [SI: rad⋅s−1] for rotational motion (as in θ=ωt) versus Ω [SI: s−1] for simple harmonic motion [as in x=xm cos(Ωt)], (b) geometric trigonometric functions whose arguments are angles [SI: rad] versus mathematical trigonometric functions whose arguments are pure numbers, (c) simple harmonic motion versus uniform circular motion in the reference circle analogy.