How can I show that the formula for the volume of a pyramid with height $h$ and an equilateral triangle as a base with side length $a$ is given by $$v =\frac{\sqrt3a^2h}{12}\;?$$
I have been poking at this question for the past few days and have not much to show for it.
Best Answer
Hint: It is convenient to put the pyramid on its side, with the apex at the origin. By using similar triangles you can find an expression for the area of cross-section of the pyramid "at" $x$. Then you can use the usual formula for the volume of a solid when areas of cross-section are known.
Added: At some stage you will want to know the area of an equilateral triangle with side $s$. By using "special angles" (and in other ways) it can be shown that this area is $\dfrac{s^2\sqrt{3}}{4}$.