geometry
So I already know how to calculate base area given all 3 edges. I need to use Herom Formula. Now I'm left with finding the height of the prism. What formula should I apply here to find the height of the prism given that I already have surface area of 1620cm (Sqr) and base area of 360cm(sqr) ? Pic of the prism:
The existing two answers are both pretty good, so here's my two cents in the form of a proof without words . . . Okay, a few words won't hurt :)
Let $h_1$ be the smallest height (up to purple), $h_3$ be the largest height to the top, and $h_2$ be the intermediate height (up to teal).
$$\mathcal{V}_{\text{whole}} = \mathcal{V}_{\text{purple}} + \mathcal{V}_{\text{teal}} + \mathcal{V}_{\text{top}} $$
The area of the base triangle is denoted $A$ as in the OP.
\begin{align*} \mathcal{V}_{\text{purple}} &= h_1 A & &\text{as a straight up prism} \\ \mathcal{V}_{\text{teal}} &= \frac23 (h_2 - h_1) A & &\text{as a straight up prism with height $h_2-h_1$ minus} \\ &&&\text{a triangular cone upside-down of the same height} \\ \mathcal{V}_{\text{top}} &= \frac13 (h_3 - h_2) A & &\text{as a triangular cone which base is still $A$ by Cavalieri principle} \end{align*} The coefficient associated with each height is easily seen to be all $1/3.\quad$Q.E.D.
To get the area of a parallelogram where you are given the side lengths and the length of one diagonal, consider a triangle with those three sides. Such a triangle is exactly half of the parallelogram.
You can then use Heron's formula to calculate the area of the triangle, and double it to get the area of the parallelogram.
Best Answer
Express the surface area by the height of the prism to have an equation. Note, for example, that the quadrilateral $A_1B_1BA$ is a rectangle (which I assume from the image).