Relations between Plane Angles and Solid Angles in a Trihedron

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, .


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[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.
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