Tetrahedron (EntityTopic, 18)

From Hi.gher. Space

Latest revision as of 14:05, 26 March 2017

A tetrahedron is the three-dimensional simplex. It is a special case of a pyramid where the base is a triangle. it is also the 3-D demicube. It is one of the five Platonic solids, containing four triangles joined three to a vertex.

Cartesian coordinates

A regular tetrahedron with edge length 2, centered at the origin, can be defined using the coordinates:

(√2/2, √2/2, √2/2);
(−√2/2, −√2/2, √2/2);
(−√2/2, √2/2, −√2/2);
(√2/2, −√2/2, −√2/2).

Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2 can be defined using the coordinates:

(-1, -√3/3, -√6/6)
(1, -√3/3, -√6/6)
(0, 2√3/3, -√6/6)
(0, 0, √6/2)

Finally, a regular tetrahedron with edge length 1 and two opposite edges parallel to the axes can be defined using the coordinates:

(¹∕₂, ^√2∕₄, 0);
(−¹∕₂, ^√2∕₄, 0);
(0, −^√2∕₄, ¹∕₂);
(0, −^√2∕₄, −¹∕₂).

• The cross sections of a tetrahedron parallel to an axis are a point that expands into a triangle.

Equations

• The hypervolumes of a tetrahedron with side length l are given by:

total edge length = 6l
surface area = √3 · l²
volume = ^√2∕₁₂ · l³

• The perpendicular height h of a tetrahedron with side length l is given by:

h = ^√6∕₃ · l

Incidence matrix