Tetrahedra with congruent face pairs
Daniel A. Klain
TL;DR
This paper proves that tetrahedra with pairs of faces of equal area must have congruent faces and derives a Heron-style volume formula for such symmetric tetrahedra.
Contribution
It establishes a geometric property linking face area equality to congruence and introduces a new volume formula for these symmetric tetrahedra.
Findings
Pairs of faces with equal area are congruent in such tetrahedra
A Heron-style volume formula is derived for symmetric tetrahedra
The result connects face area symmetry with face congruence
Abstract
If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron having this kind of symmetry.
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Taxonomy
TopicsMathematics and Applications · Advanced Theoretical and Applied Studies in Material Sciences and Geometry