
Journal for Geometry and Graphics 25 (2021), No. 2, 243251 Copyright Heldermann Verlag 2021 Angles of Isosceles Tetrahedra Hidefumi Katsuura San Jose State University, San Jose, U.S.A. hidefumi.katsuura@sjsu.edu We give three new characteristics of an isosceles tetrahedron. These three characteristics are: (i) the sum of cosines of dihedral angles of a tetrahedron at each vertex is 1, (ii) the opposing dihedral angles of a tetrahedron are pairwise same, and (iii) all four solid angles of a tetrahedron are the same. It is known that "isosceles" implies (ii) and (iii), but we think the converse of these and (i) are new. The statement (iii) suggests that a solid angle may determine an isosceles tetrahedron uniquely up to a similarity. However, we give an example to show that this is not the case unless it is a regular tetrahedron. And finally, we obtain a trigonometric identity from an isosceles tetrahedron. We use a theorem on a spherical triangle. Keywords: Isosceles tetrahedron, equifacial tetrahedron, dihedral angle, solid angle. MSC: 51M04. [ Fulltextpdf (310 KB)] for subscribers only. 