General Tetrahedron Calculator

Name: General Tetrahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a general tetrahedron. A tetrahedron is a polyhedron of four triangular faces. Mostly, the term tetrahedron is used for a regular tetrahedron, but here the general tetrahedron with different side lengths can be calculated. Its faces are the triangles (a, b, c), (a, b', c'), (a', b, c') and (a', b', c). The volume formula of the general tetrahedron was discovered by Leonhard Euler.
Enter the six side lengths and choose the number of decimal places. Then click Calculate. In every triangle, any two sides together must be longer than the third side.

Formulas:

Perimeters of the triangles:

p = a + b + c

q = a + b' + c'

r = a' + b + c'

s = a' + b' + c

A = \sqrt{\frac{p}{2} \cdot (\frac{p}{2} - a) \cdot (\frac{p}{2} - b) \cdot (\frac{p}{2} - c)} + \sqrt{\frac{q}{2} \cdot (\frac{q}{2} - a) \cdot (\frac{q}{2} - b') \cdot (\frac{q}{2} - c')} + \sqrt{\frac{r}{2} \cdot (\frac{r}{2} - a') \cdot (\frac{r}{2} - b) \cdot (\frac{r}{2} - c')} + \sqrt{\frac{s}{2} \cdot (\frac{s}{2} - a') \cdot (\frac{s}{2} - b') \cdot (\frac{s}{2} - c)}

fa = b^{2} + {b'}^{2} + c^{2} + {c'}^{2} - a^{2} - {a'}^{2}

fb = a^{2} + {a'}^{2} + c^{2} + {c'}^{2} - b^{2} - {b'}^{2}

fc = a^{2} + {a'}^{2} + b^{2} + {b'}^{2} - c^{2} - {c'}^{2}

δ = a^{2} \cdot b^{2} \cdot c^{2} + a^{2} \cdot {b'}^{2} \cdot {c'}^{2} + {a'}^{2} \cdot b^{2} \cdot {c'}^{2} + {a'}^{2} \cdot {b'}^{2} \cdot c^{2}

V = \frac{\sqrt{a^{2} \cdot {a'}^{2} \cdot fa + b^{2} \cdot {b'}^{2} \cdot fb + c^{2} \cdot {c'}^{2} \cdot fc - δ}}{12}

The lengths have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The word tetrahedron comes from ancient Greek and means four-sided figure. Tetrahedrons are the simplest polyhedra, and polyhedron, translated, means many-sided figure. Even though in most cases the regular tetrahedron is meant, which is the simplest Platonic solid, the general tetrahedron also occurs. However, its calculation is very complicated.
Besides the regular tetrahedron, there are other special cases of the general tetrahedron. First, there is the regular triangular pyramid with three equal sides, which are isosceles triangles and an equilateral triangle as a base. Then there is the corner of a cuboid with three right angles at one vertex, thus three different right triangles. Last but not least, the disphenoid is a tetrahedron with four equal, but irregular, triangles as sides.

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