How do I find the lateral area of a cuboid given its height, base area and the area of the diagonal cross-section?

Question

H, M and B are given and I need to find the lateral area (area of all the sides): Sketch of the cuboid

Since it's a cuboid, I know that the lateral area is $$S = 2(aH + bH) = 2H (a+b)$$

I found the base diagonals using the cross section area and the height. I know that they bisect each other and that they are equal in measure, but I can't find the length and breadth because i don't have any other angles.

And I can't use the fact that $d = \sqrt{a^2 + b^2}$ since $(a+b)^2$ is not the same as $a^2 + b^2$

Any kind of help is appreciated.

Answer 1

Now, $$a^2+b^2=\frac{M^2}{H^2}$$ and $$ab=B,$$ which gives $$(a+b)^2=\frac{M^2}{H^2}+2B.$$ Can you end it now?

I got $$2\sqrt{M^2+2BH^2}.$$