Tetrahedron (EntityTopic, 18)

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A tetrahedron is the three-dimensional simplex. It is a special case of a pyramid where the base is a triangle.

Cartesian coordinates

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

(1, 1, 1);
(−1, −1, 1);
(−1, 1, −1);
(1, −1, −1).

Alternatively, a regular tetrahedron with symmetry through the z-axis and edge length 2√63 can be defined using the coordinates:

(0, 0, 1);
(2√23, 0, –1/3);
(−√23, √63, –1/3);
(−√23, −√63, –1/3).

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

(12, √24, 0);
(−12, √24, 0);
(0, −√24, 12);
(0, −√24, −12).

Equations

  • The hypervolumes of a tetrahedron with side length l are given by:
total edge length = 6l
surface area = √3 · l2
volume = √212 · l3
  • The perpendicular height h of a tetrahedron with side length l is given by:
h = √63 · l

Incidence matrix

Dual: Self-dual

#TXIDVaEa3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = base of pyramid: triangle ;
3 C1a 464 = tetrahedron ;

Usage as facets


Simplices
triangletetrahedronpyrochoronpyroteronpyropeton


Demihypercubes
tetrahedronaerochorondemipenteractdemihexeract


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


10. 111
Triangular prism
11. 12
Tetrahedron
12. 4
Glome
List of tapertopes