# A rational trigonometric relationship between the dihedral angles of a tetrahedron and its circumradius

## TL;DR

This paper generalizes a known relationship between the dihedral angles and circumradius of a tetrahedron to a broader algebraic setting using rational trigonometry, applicable over various fields and geometries.

## Contribution

It extends classical tetrahedral relationships to affine spaces over arbitrary fields with rational trigonometry, broadening their applicability.

## Key findings

- Generalizes dihedral angle-circumradius relationship to affine spaces
- Uses rational trigonometry framework for broader geometric contexts
- Applies to arbitrary geometries with non-degenerate symmetric bilinear forms

## Abstract

This paper will extend a known relationship between the circumradius and dihedral angles of a tetrahedron in three-dimensional Euclidean space to three-dimensional affine space over a general field not of characteristic two, using only the framework of rational trigonometry devised by Wildberger. In this framework, a linear algebraic view of trigonometry is presented, which allows the associated three-dimensional vector space of such a three-dimensional affine space to be equipped with a non-degenerate symmetric bilinear form; this will also generalise the results presented to arbitrary geometries parameterised by such a non-degenerate symmetric bilinear form.