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A tetrahedron has four vertices as much as a triangle has three vertices. A tetrahedron therefore can have four solid angles as much as a triangle can have three angles.

I am wondering:

Is there a general formula for the solid angle at each vertex of tetrahedron, if the length of each of the six edges are given? As much as one can use the law of cosines to determine the angle at each vertex of a triangle, as long as the lengths of each sides of triangle is given?

Further question: what kind of area in mathematics does such a theorem belong to? What is the most general textbook for such mathematics?

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If you have all of the edge lengths, you can compute the corner angles for each face. The solid angle at each vertex is then the area of a spherical triangle whose sides are the corner angles of the three faces that meet. –  Henning Makholm Oct 14 '11 at 12:16
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On that note, there's l'Huilier's theorem... see also this paper. –  J. M. Oct 14 '11 at 12:43
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See also en.wikipedia.org/wiki/Solid_angle#Tetrahedron –  joriki Oct 14 '11 at 13:15

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