This question is inspired by the question The volume for truncated pyramid with irregular base.
Given that we have the top and bottom surface area ($A_1$, $A_2$) of a pyramid, ad the height of the truncated pyramid is given by $H$, how can we find the surface area of the pyramid, given the height from the bottom $h$? Without a loss of generality we can assume that $A_2$ is always bigger than $A_1$.
I can prove that in two limiting cases where the surface is circle and rectangle, the area $A_h$ at height $h$ is somehow proportional to the square of $h$, ie: $A_h
\varpropto h^2$. But is there a general formula connecting $A_h$, $h$, $H$, $A_1$ and $A_2$?
Best Answer
$A_h$ is proportional with the square of the distance from its plane to the virtual top of the pyramid at height $H_{top}$, so from $(\frac{H_{top}-H}{H_{top}})^2=\frac{A_1}{A_2}$ it follows that $H_{top}=\frac H{1-\sqrt{\frac{A_1}{A_2}}}$ and $A_h=(1-\frac h{H_{top}})^2A_2$.