Circumradius of a tetrahedron

Question

I put the following formula for the circumradius of a tetrahedron on the Wikipedia page on the tetrahedron, but it was deleted for lack of a citation. Does anyone have a reference for it?

Here, $a,b,c$ are three edges that meet at a point; $A,B,C$ are the opposite edges; and $V$ is the volume of the tetrahedron. $$R = \frac{\sqrt{(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}$$

I happen to use the formula in my note "Heron-like Hedronometric Results for Tetrahedral Volume" (PDF) (the very last line!), but I don't provide a justification for it. (As I recall, I derived it from MathWorld's coordinate-based formula via a bunch of symbol-crunching with Mathematica.) — Blue, Jun 15 '18 at 03:41
How about publishing this as a note in some expository journal, then citing that? — Angina Seng, Jun 15 '18 at 04:06

achille hui · Accepted Answer · 2018-06-19T05:59:12.237

5

This is a result by A. L. Crelle. It first? appeared in

Crelle, A. L. "Einige Bemerkungen über die dreiseitige Pyramide." Sammlung mathematischer Aufsätze u. Bemerkungen 1, 105-132, 1821.
(~~an online copy is available on archive.org~~)

The statement appears at page 117 (item 186) and the proof is one or two pages around that. Since the article is in German and I don't speak German, I don't know where the proof exactly starts.

Another English reference is

I. Todhunter, "Spherical Trigonometry: For the Use of Colleges and Schools" (1886)
(online copies are available under Project Gutenberg 1 and archive.org 2, 3, 4 ).

The material is covered at page 129 ( Art. 163 ).

edited Jun 19 '18 at 05:59

answered Jun 15 '18 at 12:47

achille hui

122,701

Thanks for that. – Empy2 Jun 15 '18 at 12:51
There are also several copies of Todhunter's book on archive.org: 1, 2, 3. – Jean-Claude Arbaut Jun 19 '18 at 05:48
@Jean-ClaudeArbaut Thanks for the links. Since Todhunter's book is also available on amazon. In the worst case, I can always point it to that. – achille hui Jun 19 '18 at 05:53
1

Well, Isaac Todhunter died in 1884, so I don't think there is any risk of copyright infringement (I would not have given the links otherwise). – Jean-Claude Arbaut Jun 19 '18 at 06:07

score 2 · Answer 2 · answered Jun 15 '18 at 05:49

In "An Open Problem on Metric Invariants of Tetrahedra" (citation below), Lu Yang and Zhenbing Zeng mention matter-of-factly that the circumradius $R$ satisfies

$$R^2 = -\frac{M_5}{2 M_0}$$

where $M_0$ is the Cayley-Menger determinant, and where $M_5$ is the principal minor determinant obtained by deleting the $5$th row and column of $M_0$. That is, in the notation of OP, $$\begin{align} M_0 := \left|\begin{array}{ccccc} 0 & a^2 & b^2 & c^2 & 1 \\ a^2 & 0 & C^2 & B^2 & 1 \\ b^2 & C^2 & 0 & A^2 & 1 \\ c^2 & B^2 & A^2 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{array}\right| &= 288 V^2 \\[8pt] M_5 := \left|\begin{array}{cccc} 0 & a^2 & b^2 & c^2 \\ a^2 & 0 & C^2 & B^2 \\ b^2 & C^2 & 0 & A^2 \\ c^2 & B^2 & A^2 & 0 \\ \end{array}\right| &= \begin{array}{c} -(a A + b B + c C)(-a A + b B + c C) \\ \cdot(a A - b B + c C)(a A + b B - c C) \end{array} \end{align}$$ which gives the relation OP mentions.

I suspect that the relation should appear in a comprehensive reference of Cayley-Menger lore, but my cursory web search has been fruitless.

Here's the citation information I have for the Yang-Zeng note:

Lu Yang and Zhenbing Zeng, An open problem on metric invariants of tetrahedra, Proceedings of the 2005 international symposium on Symbolic and algebraic computation (ISSAC ’05). ACM, New York, NY, USA, 362-364. DOI=10.1145/1073884.1073934 http://doi.acm.org/10.1145/1073884.1073934

A PDF of the paper containing the note appears to be available from ResearchGate: https://www.researchgate.net/publication/221563984_An_open_problem_on_metric_invariants_of_tetrahedra

Thanks. I'll leave it for a day or two in case there is a more specific reference to it. — Empy2, Jun 15 '18 at 06:34

Circumradius of a tetrahedron

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