Supplementary Solid Angles for Trihedron

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration constructs a supplementary solid angle for a given trihedral solid angle. Let , and be the edges of a trihedron that determines the solid angle. The plane angles opposite the edges are denoted , , and the dihedral angles at the edges are denoted , , . Let be a point inside the trihedron and denote its orthogonal projections onto the faces of the trihedron by , and . Then , and are edges of a trihedron that determines the supplementary space angle.


The plane angles of the supplementary angle are , and , and its dihedral angles are , and .

The measure of the initial trihedral angle is (the spherical excess formula for a trihedron), while the measure of its supplementary angle is .


Contributed by: Izidor Hafner (March 2017)
Open content licensed under CC BY-NC-SA



This Demonstration gives an animation for Figure 5.5 in [3, p. 186].

The deficiency of a solid angle determined by an -sided spherical polygon with angles , , …, is . The deficiency of a solid angle equals its supplementary angle [3, pp. 186–187].


[1] Wikipedia. "Spherical Law of Cosines." (Feb 23, 2017) _cosines.

[2] Wikipedia. "Spherical Trigonometry." (Feb 23, 2017)

[3] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1997.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.