Lengths edges of a tetrahedron

a1 =
a2 =
a3 =
a4 =
a5 =
a6 =

 All results

 Square volume is positive
(tetrahedron exists)

 Square volume is equal to zero
(tetrahedron does not exist; all points are in one plane)

 Square volume is negative
(tetrahedron does not exist)

Ukr   Rus   Eng
v - The volume of a tetrahedron
v2 - Square volume of a tetrahedron

Kuzmich V.I., Kuzmich Y.V.
Kherson State University

Consider the problem of finding volume of a tetrahedron (triangular pyramid), if you know the lengths of all its edges. This problem was solved by Joachim Jungius (1587-1657), and therefore set them formula for the volume of a tetrahedron is called Yungius formula [1, p. 100]. This formula is quite complicated, so to record it, we introduce the following notation edges of the tetrahedron.

Given a tetrahedron SABC (figure 1), the length of its edges is denoted by: , and its volume by – .

Fig. 1. Tetrahedron.

With this notations, the formula Yungius will look like:

According to this formula, you can find the volume of the tetrahedron by the lengths of its edges. However, not all six segments there will be a tetrahedron, for which these segments are the edges. Calculations show that for the same set of six segments may exist, such permutation in which the edges of a tetrahedron with the data will exist, while other permutations – will not exist. Consequently, the volume of the tetrahedron depends on its orientation.

For the six preset intervals of 720 possible different permutations. Therefore, to establish the orientation of the tetrahedron at which it can exist is a difficult task, with many calculations. This process is simplified by using a special calculator developed by the authors, and which is programmed to calculate the cyclic right side of Yungius. The disadvantage of this method is the computational error, so if the result is close to zero should improve their accuracy.

To calculate the volume of a tetrahedron is enough to enter on the left side of the working field calculator in the appropriate fields lengths of the edges of the tetrahedron: , and then select the desired set of values of the square (all values are positive, zero or negative) and activate the button "calculate" (figure 2).

Fig. 2. The working field of the calculator.

On the right side of the working field to complete the calculations will show "The calculation is complete." and will be given all the results of calculations, showing the impossibility of the existence of a tetrahedron in the case of zero or negative value of the square, as well as specifying the number of results. The square of the volume of the tetrahedron is denoted by «v2», and the volume – «v».

The calculator provides a system message about the incorrect data entry, "Input the value of an» – in the case of unfilled fields «an», and «The value of an is incorrect (s). Use point instead of commas for floating» – in the case of fields «an» not the numerical value "s", or when used commas in the recording of a fractional number.

Calculator is presented in Ukrainian, Russian and English versions. The authors thank Boukoulou Didier Criss and Alferov E. for help in translating into English the calculator, and for adapting the calculator on the website of the Kherson State University.


1. Я.П. Понарин. Элементарная геометрия: В 2 т. — Т. 2: Стереометрия, преобразования пространства. — М.: МЦНМО, 2006.— 256 с.

Last modified: Thursday, 30 January 2014, 9:03 PM