Katsuura, Hidefumi Solid angle sum of a tetrahedron. (English) Zbl 1447.51020 J. Geom. Graph. 24, No. 1, 29-34 (2020). 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. Cited in 3 Documents MSC: 51M16 Inequalities and extremum problems in real or complex geometry 51M04 Elementary problems in Euclidean geometries Keywords:solid angles of a tetrahedron; dihedral angles; direction angles; right tetrahedron; equifacial tetrahedron Citations:Zbl 0046.37903 × Cite Format Result Cite Review PDF Full Text: Link References: [1] S. Abu-Saymeh, M. Hajja:On the Fermat-Torricelli Points of Tetrahedra and of Higher Dimensional Simplexes. Math. Mag.70(5), 372-378 (1997). · Zbl 0914.51018 [2] N. Altshiller-Court:Modern Pure Solid Geometry. The Macmillan Co., New York 1935. · JFM 61.1383.03 [3] F. Eriksson:On the Measure of Solid Angles. Math. Mag.63(3), 184-187 (1990). · Zbl 0721.51023 [4] J.W. Gaddum:The Sum of the Dihedral and Trihedral Angles in a Tetrahedron. Amer. Math. Monthly59(6), 370-371 (1952). · Zbl 0046.37903 [5] H. Katsuura:Three-Dimensional Viviani Theorem on a Tetrahedron. J. Geometry Graphics23/2, 179-182 (2019). · Zbl 1439.51027 [6] A. Liu, B. Joe:Relationship Between Tetrahedron Shape Measures. BIT Numerical Mathematics34, 268-287 (1994). · Zbl 0806.65104 [7] J.C. Lagarius, T.J. Richardson, J.H. Lindsey:Solid Angle of a Tetrahedron. Problem Section of Amer. Math. Monthly106(3), 268-270 (1999). [8] A.V. Oosterom, J. Strackee:The Solid Angle of a Plane Triangle. IEEE Trans. Biomed. Eng. BME-30(2), 125-126 (1983). [9] K. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.