<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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 <italic>angular magnitude</italic>, defined in terms of congruence classes of angles, and the (numerical) <italic>angular measure</italic> that can be assigned to each congruence class in such a way that, e.g., the right angle has the numerical value <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_msr-2022-0019_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></math>
<tex-math>{\pi  \over 2}</tex-math>
</alternatives>
</inline-formula>. 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.</p>
</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 angular magnitude, defined in terms of congruence classes of angles, and the (numerical) angular measure that can be assigned to each congruence class in such a way that, e.g., the right angle has the numerical value 


π2
{\pi  \over 2}

. 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.

On Angular Measures in Axiomatic Euclidean Planar Geometry

NTNU Norwegian University of Science and Technology

Measurement Science Review

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On Angular Measures in Axiomatic Euclidean Planar Geometry"}

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...

On Angular Measures in Axiomatic Euclidean Planar Geometry

Pubblicato online: 14 mag 2022

Pagine: 152 - 159

Ricevuto: 19 nov 2021

Accettato: 31 mar 2022

DOI: https://doi.org/10.2478/msr-2022-0019

Parole chiave
Angle magnitude, Euclidean geometry, radians, International System of Units, SI units

© 2022 Martin Grötschel et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On Angular Measures in Axiomatic Euclidean Planar Geometry

Pubblicato online: 14 mag 2022

Pagine: 152 - 159

Ricevuto: 19 nov 2021

Accettato: 31 mar 2022

DOI: https://doi.org/10.2478/msr-2022-0019

Parole chiaveAngle magnitude, Euclidean geometry, radians, International System of Units, SI units

© 2022 Martin Grötschel et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Parole chiave
Angle magnitude, Euclidean geometry, radians, International System of Units, SI units