Question

Find the volume of the described solid $S$.
A pyramid with height $h$ and base an equilateral triangle with side $a$ (a tetrahedron).

Answer 1

Determine the volume of the pyramid:

The volume of a pyramid is $\frac{1}{3}$times the product of the base area of the pyramid and its height.

The base is an equilateral triangle.

So, the semi perimeter of the base will be $\frac{3a}{2}$.

Find the area of the equilateral triangle:

$\begin{array}{rcl}A& =& \sqrt{\frac{3}{2}a{\left(\frac{3}{2}a-a\right)}^3}\\ & \Rightarrow & A=\sqrt{\frac{3}{2}a{\left(\frac{1}{2}a\right)}^3}\\ & \Rightarrow & A=\frac{\sqrt{3}}{4}{a}^2\end{array}$

Determine the volume of the pyramid with the calculated base and the height $h$:

$\begin{array}{rcl}V& =& \frac{1}{3}\times \frac{\sqrt{3}}{4}{a}^2\times h\\ & \Rightarrow & V=\frac{\sqrt{3}}{12}{a}^{2}h\end{array}$

So, the volume is $\begin{array}{rcl}V& =& \frac{\sqrt{3}}{12}{a}^{2}h\end{array}$.