Relations between Plane Angles and Solid Angles in a Trihedron

Requires a Wolfram Notebook System

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

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Let , and be the edges of a trihedron that determines a solid angle. The plane angles opposite the edges are denoted , , , and the angles between the edges and their opposite faces are denoted , , . Construct three planes parallel to the faces , and at distance 1 from the corresponding faces. Let the intercepts of these planes with edges of the solid angle be , , . Also define the points , , , such that , , , , to get a parallelepiped with all faces of equal area, since all heights are equal. The lengths of the edges are , and . The areas of the faces are , and .


Since the areas are equal, .


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




[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.