# Tetrahedral Geometry from Areas

## TL;DR

This paper addresses a classical geometric problem by deriving a comprehensive description of a Euclidean tetrahedron's geometry solely from face and medial parallelogram areas, with implications for quantum gravity.

## Contribution

It introduces new formulas for dihedral angles, face angles, and edge lengths based on face and medial parallelogram areas, expanding classical tetrahedral geometry results.

## Key findings

- Derived expressions for dihedral angles and edge lengths from areas
- Provided an alternative proof using bivectors for a known area relation
- Corrected previous errors and added historical context

## Abstract

We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles and (an) edge length, the remaining parts being derivable by symmetry or by identities in the classic 1902 compendium of results on tetrahedral geometry of Richardson. We also provide an alternative proof using (bi)vectors of the result of Yetter that four times the sum of the squared areas of the medial parallelograms is equal to the sum of the squared areas of the faces. The updated version corrects several errors and provides better historical context for the result.