# 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 $2\mathrm{arctan}\sqrt{2}$ ($\approx {109}^{\circ}$), 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 $\frac{a\sqrt{6}}{3}$.

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 $\frac{{a}^{3}\sqrt{2}}{12}$.

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 |