Tetrahedron: Surface area and volume

The tetrahedron is a figure formed by $4$ equilateral triangles.

To calculate the area of the tetrahedron: $$A_{tetrahedron}=4\cdot {\displaystyle \frac{a\cdot h}{2}}=2\cdot a\cdot h$$

The triangles that compose the tetrahedron are equilateral, so it is possible to express its height $h$ according to its base: $$\begin{gathered} a^2=({\displaystyle \frac{a}{2}}{)}^2+h^2 \\ {h}^2=a^2({\displaystyle \frac{3}{4}}) \\ h=a\cdot {\displaystyle \frac{\sqrt{3}}{2}} \end{gathered}$$

And the area of the tetrahedron is: $$A_{tetrahedron}={\displaystyle \frac{a\cdot a\cdot {\displaystyle \frac{\sqrt{3}}{2}}}{2}}={\displaystyle \frac{a^2\cdot \sqrt{3}}{4}}=a^2\cdot \sqrt{3}$$

Finally, the expression of the volume of the regular tetrahedron is: $$V={\displaystyle \frac{\sqrt{2}}{12}}\cdot a^3$$