properties of regular tetrahedron

A regular tetrahedronMathworldPlanetmathPlanetmathPlanetmath may be formed such that each of its edges is a diagonal of a face of a cube; then the tetrahedronMathworldPlanetmath has been inscribedMathworldPlanetmath in the cube.

It’s apparent that a plane passing through the midpointsMathworldPlanetmathPlanetmathPlanetmath of three parallel edges of the cube cuts the regular tetrahedron into two congruentPlanetmathPlanetmath pentahedronsMathworldPlanetmath and that the intersection figure is a square, the midpoint M of which is the centroid of the tetrahedron.

The angles between the four half-lines from the centroid M of the regular tetrahedron to the vertices ( are 2arctan2 (109), which is equal the angle between the four covalent bonds of a carbon .  A half of this angle, α, can be found from the right triangleMathworldPlanetmath in the below figure, where the catheti are s2 and s2.


One can consider the regular tetrahedron as a cone.  Let its edge be a and its height h.  Because of symmetryMathworldPlanetmathPlanetmath, a height line intersects the corresponding base triangle in the centroid of this equilateral triangleMathworldPlanetmath.  Thus we have (see the below ) the rectangular triangle with hypotenuseMathworldPlanetmath a, one cathetus h and the other cathetus (  23a32=a3  (i.e. 23 of the median ( a32 of the equilateral triangle — see the common point of triangle medians).  The Pythagorean theoremMathworldPlanetmathPlanetmath then gives


Consequently, the height of the regular tetrahedron is a63.

Since the area of the base triangle ( is a234, the volume (one third of the product of the base and the height) of the regular tetrahedron is a3212.

Title properties of regular tetrahedron
Canonical name PropertiesOfRegularTetrahedron
Date of creation 2013-03-22 18:29:39
Last modified on 2013-03-22 18:29:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Topic
Classification msc 51E99
Synonym regular tetrahedron
Related topic Grafix
Related topic DehnsTheorem
Related topic Tetrahedron