properties of regular tetrahedron

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

It’s apparent that a plane passing through the midpoints of three parallel edges of the cube cuts the regular tetrahedron into two congruent pentahedrons 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 (http://planetmath.org/Polyhedron) are $2\arctan \sqrt{2}$ ($\approx {109}^{\circ }$), which is equal the angle between the four covalent bonds of a carbon .  A half of this angle, $\alpha$, can be found from the right triangle in the below figure, where the catheti are $\frac{s}{\sqrt{2}}$ and $\frac{s}{2}$.

One can consider the regular tetrahedron as a cone.  Let its edge be $a$ and its height $h$.  Because of symmetry, a height line intersects the corresponding base triangle in the centroid of this equilateral triangle.  Thus we have (see the below ) the rectangular triangle with hypotenuse $a$, one cathetus $h$ and the other cathetus (http://planetmath.org/Cathetus)  $\frac{2}{3}\cdot \frac{a\sqrt{3}}{2}=\frac{a}{\sqrt{3}}$  (i.e. $\frac{2}{3}$ of the median (http://planetmath.org/Median) $\frac{a\sqrt{3}}{2}$ of the equilateral triangle — see the common point of triangle medians).  The Pythagorean theorem then gives

$$h=\sqrt{a^2-{\left(\frac{a}{\sqrt{3}}\right)}^2}=\frac{a\sqrt{6}}{3}.$$

Consequently, the height of the regular tetrahedron is ${\displaystyle \frac{a\sqrt{6}}{3}}$.

Since the area of the base triangle (http://planetmath.org/EquilateralTriangle) is $\frac{a^2\sqrt{3}}{4}$, the volume (one third of the product of the base and the height) of the regular tetrahedron is ${\displaystyle \frac{a^3\sqrt{2}}{12}}$.

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