Solid angles in perspective

TL;DR

This paper clarifies the concept of solid angles, explores their units and interrelations, and demonstrates their educational value in introducing advanced geometric topics to students.

Contribution

It introduces the solid degree as a new unit for solid angles and discusses their applications in various geometric and scientific contexts.

Findings

Solid angles can be expressed in units like steradians, square degrees, and a new solid degree.

Using solid angles aids in understanding crystal structures, spherical trigonometry, and non-Euclidean geometry.

The paper promotes solid angles as accessible tools for teaching advanced geometry.

Abstract

The specialised uses of solid angles mean that they are quite unfamiliar quantities. This article, apart from making solid angles a little more familiar, brings out several topics of general interest, such as how units are interrelated and how equations depend on the choice of units. Although the steradian is commonly used as the unit for solid angle, another unit, the square degree, is used in astronomy, and a unit introduced here, the solid degree (with 360 solid degrees in a hemisphere) could be used with benefits that are similar to those of the degree when it is used as the unit for plane angle. The article, which is suitable for students at A-level and introductory undergraduate level, also shows how solid angles can provide a gentle introduction to crystal structure, spherical trigonometry and non-Euclidean geometry.