Journal for Geometry and Graphics 24 (2020), No. 1, 029--034
Copyright Heldermann Verlag 2020
Solid Angle Sum of a Tetrahedron
Hidefumi Katsuura
Dept. of Mathematics and Statistics, San Jose State University, One Washington Square, San Jose, CA 95192-0103, U.S.A.
hidefumi.katsuura@sjsu.edu
J. W. Gaddum proved in 1952 that the solid angles sum of a tetrahedron is less than 2π 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 will give a new proof of this result by embedding the tetrahedron into a parallelepiped. In addition, we will 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.
Keywords: solid angles of a tetrahedron, dihedral angles, direction angles, right tetrahedron, equifacial tetrahedron.
MSC: 51M16; 51M04.
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