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The tetrahedron (of the Greek will tétra : four), is a solid made up of four Triangle S, of the family of the Pyramide S, therefore cone S.

The regular tetrahedron, formed of four equilateral triangles, fact part of the five polyhedral regular, or solid of Plato.

Orthocentric tetrahedron: a tetrahedron which has its 4 convergent heights is known as orthocentric. The point of contest is then the orthocentre of the tetrahedron.

The tetrahedron is a Simplexe of degree 3.

The volume of a tetrahedron is equal to \ scriptstyle V= {} ^1 \! /\! _3 Bh if B is the surface of a base of the tetrahedron and h the height of the tetrahedron being based on this base.

Regular tetrahedron

If a is the length of an edge:
  • surface is equal to: A= \ sqrt {3} a^2
  • the height is equal to: H= {\ scriptstyle \ sqrt {\ frac23}} a
  • the center of the tetrahedron is located, compared to the base, with: h= \ tfrac14 H
  • volume is equal: V= \ tfrac {1} {12} \ sqrt {2} a^3
  • the value of the cosine of the central angle of the regular tetrahedron (i.e. that which all the segments form which leave the center towards the four tops) is of - ⅓.

The tetrahedron is its clean Dual, i.e. by uniting the centers of the faces of a regular tetrahedron, one obtains a new regular tetrahedron.

The group of the Isométrie S leaving overall invariant the regular tetrahedron is isomorphous with the symmetrical Groupe \ mathfrak {S} _4

Simple: Tetrahedron

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