2017 Mathcounts State Prep: Volume of a Regular Tetrahedron and Its Relationship with the Cube it's Embedded

How to find volume of a tetrahedron (right pyramid) with side length one.

The above link gives you a visual interpretation of the relationship of a regular tetrahedron, its
relationship with the cube that it is embedded and the other kind of tetrahedron (right angle pyramid).

The side of the cube is $\dfrac {S} {\sqrt {2}}$ so the volume of the regular embedded tetrahedron is
$\dfrac {1} {3}\times \left( \dfrac {S} {\sqrt {2}}\right) ^{3}$=$\dfrac {1} {3}*\dfrac {s^{3}} {2\sqrt {2}}$= $\dfrac {\sqrt {2}S^{3}} {12}$.

You can also fine the height of the tetrahedron and then $\dfrac {1} {3}$*base*space height to get the volume.
Using Pythagorean theory, the hypotenuse S and one leg which is $\dfrac {2} {3}$ of the height of the equilateral triangle base, you'll get the space height.