Regular tetrahedron/Introductory problem

Problem

Find the volume of a tetrahedron whose sides all have length $2$ .

We find the area of the base:

$\mathrm {A} =\dfrac{4\sqrt{3}}{4}=\sqrt{3}$

Now we find the lateral height:

$\mathrm {lh}=\sqrt{2^2-1^2}=\sqrt{3}$

Now we can find the height of the tetrahedron:

$\mathrm {h}=\sqrt{\sqrt{3})^2-(\dfrac{1}{3}*\sqrt{3})^2}=\sqrt{\dfrac{8}{3}$

Now the volume of the tetrahedron is:

$\dfrac{1}{3}*\sqrt{3}*\sqrt{\dfrac{8}{3}}=\dfrac{\sqrt{8}}{3}$