Theory of Parallels 27: Solid Angles

Summary

A trihedral angle equals half the sum of its dihedral angles minus a right angle.

Measuring Solid Angles

A solid angle is dihedral if it is bounded by two planes meeting at a line; trihedral if it is bounded by three planes at a point. Thus, a tetrahedron contains four trihedral angles, one at each vertex, and six dihedral angles, one at each edge. TP 27 relates the measure of a trihedral angle to the measures of the three dihedral angles at its edges. We know how to measure dihedral angles (see the TP 26 notes), but how does one measure a trihedral angle? We must answer this question before we can understand the statement of TP 27, much less prove it.

We tend to associate angle measurement with rotation. The measure of an ordinary angle in the plane, for example, indicates the amount of rotation required to bring one of its arms into coincidence with the other. Similarly, the measure of a dihedral angle indicates the amount of rotation required to bring one of its faces into coincidence with the other. The link between angle measure and rotation, however, ceases to exist when we work with trihedral angles (or more generally, when we work with polyhedral angles, formed by three or more planes meeting a point). Fortunately, we can articulate a general definition of polyhedral angle that agrees with our existing measures, but that is not based on rotation.

To motivate this definition, let us examine an alternate, protractor-free method for measuring ordinary angles in the plane. We begin by assigning a numerical value to the “full angle” (a 360° rotation).