Disphenoid Calculator

Name: Disphenoid - Geometry Calculator
Author: Jürgen Kummer

Calculations at a disphenoid or isosceles tetrahedron. A disphenoid is a tetrahedron with four congruent triangles as faces. Its opposite edges have the same length.
Enter the three lengths a, b and c and choose the number of decimal places. Then click Calculate. The squares of any two side lengths must be larger than the square of the third side length (e.g. a=4, b=5, c=6).

Formulas:

Perimeter of one triangle:

p = a + b + c

A = 4 \cdot \sqrt{\frac{p}{2} \cdot (\frac{p}{2} - a) \cdot (\frac{p}{2} - b) \cdot (\frac{p}{2} - c)}

V = \sqrt{\frac{(a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (- a^{2} + b^{2} + c^{2})}{72}}

r_{c} = \sqrt{\frac{a^{2} + b^{2} + c^{2}}{8}}

r_{i} = \frac{3 \cdot V}{A}

Lengths and radiuses 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.

Sphenoid is ancient Greek for wedge, but this term doesn't refer to the specific geometric shape of a wedge, but rather to an open shape consisting of two faces, a so-called dihedron. These faces are two congruent triangles. A disphenoid is composed of two such sphenoids. The prefix di means two.
If the faces of the disphenoid are isosceles triangles, then it is a tetragonal disphenoid. If they are equilateral triangles, then it is a regular tetrahedron. The general disphenoid is also called a rhombic disphenoid, because if one of the two dihedrons is unfolded into a plane, it forms a rhombus.

The general disphenoid is rotationally symmetric about three axes at an angle of 180 degrees and multiples thereof. These rotation axes pass through the midpoints of a pair of opposite edges. It is neither mirror-symmetric nor point-symmetric. Tetrahedra always have four vertices, six edges, and four faces. This also applies to the general and the tetragonal disphenoid.

Disphenoids occur in nature as crystal forms and as atomic structures in minerals. For example, in the mineral zircon, the SiO₄ tetrahedron is slightly distorted and forms a tetragonal disphenoid instead of a regular tetrahedron.

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