Question

What is the volume of a pyramid with an equilateral triangle for a base and a height equal to the base side length? Express your answer in terms of s, the length of a side of the base

Answer 1

$\bf \begin{array}{llll} \textit{area of an equilateral triangle}\\\\ A=\cfrac{s^2\sqrt{3}}{4}\qquad \begin{cases} s=length~of\\ \qquad a~side \end{cases} \end{array}\qquad \begin{array}{llll} \textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3}~~ \begin{cases} B=area~of\\ \qquad its~base\\ h=height\\[-0.5em] \hrulefill\\ B=\cfrac{s^2\sqrt{3}}{4}\\[1em] h=s \end{cases} \end{array}$

$\bf V=\cfrac{~~\left( \frac{s^2\sqrt{3}}{4} \right)(s)~~}{3}\implies V=\cfrac{~~\frac{s^3\sqrt{3}}{4}~~}{\frac{3}{1}} \\\\\\ V=\cfrac{s^3\sqrt{3}}{4}\cdot \cfrac{1}{3}\implies V=\cfrac{s^3\sqrt{3}}{12}$