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Solid angle sum of a tetrahedron. (English) Zbl 1447.51020

Summary: J. W. Gaddum proved in [Am. Math. Mon. 59, 370–371 (1952; Zbl 0046.37903)] that the solid angles sum of a tetrahedron is less than \(2\pi\) by finding the bound to the sum of six angles between four vertical segments from an interior point to the faces of the tetrahedron. We give a new proof of this result by embedding the tetrahedron into a parallelepiped. In addition, we give the bound on the sum of the four solid angles of a right tetrahedron using direction angles, and prove that the sum of the four solid angles of an equifacial tetrahedron is at most that of a regular tetrahedron.

MSC:

51M16 Inequalities and extremum problems in real or complex geometry
51M04 Elementary problems in Euclidean geometries

Citations:

Zbl 0046.37903

References:

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