On angular measures in axiomatic Euclidean planar geometry

TL;DR

This paper provides a rigorous axiomatic foundation for angular measure in Euclidean geometry, clarifying its distinction from length and reaffirming the radian as a fixed, unalterable unit in SI.

Contribution

It offers a formal axiomatic presentation of angular measure, emphasizing its intrinsic difference from length and supporting the current SI definition of the radian.

Findings

Angular measure is a scalar quantity derived from arc length and radius.

Angles are fundamentally different from lengths, with no natural units like the metre.

The SI definition of the radian should remain unchanged due to its mathematical nature.

Abstract

We address the issue of angular measure, which is a contested issue for the International System of Units (SI). We provide a mathematically rigorous and axiomatic presentation of angular measure that leads to the traditional way of measuring a plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc, a scalar quantity. We distinguish between the \emph{angular magnitude}, defined in terms of congruence classes of angles, and the (numerical) \emph{angular measure} that can be assigned to each congruence class in such a way that, e.g., the right angle has the numerical value $\frac\pi2$. We argue that angles are intrinsically different from lengths, as there are angles of special significance (such as the right angle, or the straight angle), while there is no distinguished length in Euclidean geometry. This is further underlined by the observation that, while units such as the metre and kilogram have been refined over time due to advances in metrology, no such refinement of the radian is conceivable. It is a mathematically defined unit, set in stone for eternity. We conclude that angular measures are numbers, and the current definition in SI should remain unaltered.