How to calculate top base area with bottom base area and height of frustum?

Question

I have the following frustum

The bottom base area $A_1$ is known, the top base area $A_2$ is unknown.

We know this about the frustum

We know the height $h$ and the angle $a$ of the frustum.

Can the top base area $A_2$ be calculated from the given facts and how?

Answer 1

You need some other information to calculate top base area.

Let $d_1$ and $d_2$ be the lengths of upper and lower base diagonals. We have $d_2=d_1-2h/\tan a$ and the bases are similar, so that: $$ {A_2\over A_1}=\left({d_2\over d_1}\right)^2= \left(1-{2h\over d_1\tan a}\right)^2. $$ As you can see, we need to know $d_1$ to compute $A_2$. But $A_1={1\over2}d_1^2\sin\theta$, where $\theta$ is the angle between the diagonals of each base. We can find $d_1$ only if we know the value of $\theta$ or some other information equivalent to it.