A four-faced (tetra-) polyhedron with each face shaped like a triangle. In a regular tetrahedron each face is an equilateral triangle.

A regular tetrahedron looks like a pyramid with 3 sides (faces) and a base rather then four.

One of the 5 platonic solids.

3 faces (and edges) meet at every vertex.

So the dual polyhedron is the tetrahedron itself!

If a tetrahedron is defined by three vectors, the volume of that tetrahedron can be found by evaluating the value of one-sixth of the scalar triple product. So, if the tetrahedron has corners at points A, B, C and D, the three vectors needed are those in the directions AB, AC and AD. Then the volume V is found by

V = (1/6)(AD.ABxAC).

As an example, let us consider a tetrahedron which is defined by the following vectors:

AB = 5i - j - k
AC = 2i - 8j + k
AD = -i + 2k

A quick way to evaluate the scalar triple product is to calculate the modulus of the determinant of a matrix consisting of these three vectors:

         |     / 5  -1  -1 \ |
V = (1/6)| det | 2  -8   1 | |
         |     \-1   0   2 / |

= (1/6)|5(-16 - 0) + 1(4 + 1) - 1(0 - 8)|
= (1/6)|-80 + 5 + 8|
= (1/6)|-67|
= (1/6)(67)
= 67/6

Therefore the tetrahedron has volume 67/6 units^3.

Tet`ra*he"dron (?), n. [Tetra- + Gr. seat, base, fr. to sit.] Geom.

A solid figure inclosed or bounded by four triangles.

In crystallography, the regular tetrahedron is regarded as the hemihedral form of the regular octahedron.

Regular tetrahedron Geom., a solid bounded by four equal equilateral triangles; one of the five regular solids.

 

© Webster 1913.

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