Angles of Isosceles Tetrahedra

Publication Date

1-1-2021

Document Type

Article

Publication Title

Journal for Geometry and Graphics

Volume

25

Issue

2

First Page

243

Last Page

251

Abstract

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

Department

Mathematics and Statistics

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