On the status of plane and solid angles in the International System of Units (SI)

Abstract

The article analyzes the arguments that became the basis for declaring in 1995, at the 20th General Conference on Weights and Measures that the plane and solid angles are dimensionless derived quantities in the International System of Units. The inconsistency of these arguments is shown. It is found that a plane angle is not a derived quantity in the SI, and its unit, the radian, is not a derived unit. A solid angle is the derived quantity of a plane angle, but not a length. Its unit, the steradian, is a coherent derived unit of the radian.

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

“…Table 2, here, would replace table 1 in the brochure. This proposal essentially follows those of Romain [34], Brownstein [28] and Kalinin [29], and is similar to that of Torrens [19] (but without the redefinitions of torque, angular momentum and moment of inertia). It is important to stress that conventional definitions of torque, angular momentum and moment of inertia are retained.…”

“…Quincey and Burrows [26], have called this dimensional constant of Nature the Cotes constant, with symbol θ N , honouring Roger Cotes, who, in 1714, introduced the concept of the 'radian' in everything but name [16]. Instead of referring to the radian directly, Kalinin [29] uses the factor Φ/(2π), where Φ represents one revolution.…”

“…With the recent redefinitions of SI units [1], defining constants consist of invariant atomic properties or important physical constants. For angle, the most direct defining constant is one revolution [29]-as compared with a right-angle [27], a straight-angle, or the radian itself, all of which require geometrical construction. A revolution is easily comprehended and requires no construction: its magnitude can be precisely determined because it involves a null measurement.…”

“…There is a strong relationship between solid angle (in terms of the base quantity angle) and area (in terms of the base quantity length) [29]. It is therefore instructive to first consider the definition of area.…”

“…In a recent article entitled 'on the status of plane and solid angles in the international system of units (SI)' [29], Mikhail Kalinin discusses arguments leading to the 1980 CIPM decision to reclassify the radian and steradian as dimensionless derived units, which implies that plane angle is supposed to be conceptualized as a dimensionless ratio of two lengths and solid angle as a dimensionless ratio of an area and a length squared. He points out that the series expansion of trigonometric functions, sometimes used to 'justify' the supposed dimensionless nature of angle, is inappropriate because of the confusion between two different kinds of trigonometric functions, where, in one case, the argument is an angle and, in the other case, the argument is a dimensionless number.…”