Scratching my rusty memories a bit, especially about the properties of radicals (!!!), I managed to find out by myself all the results that I couldn't find elsewhere online. So I have been very proud of myself after so many years out of practice... I started examining the geometry of the dome as an half-sphere of radius R, in comparison with the half-cube of height a (thus with base 2a). So the basic formulas of volumes V and areas A are the following:
Then I solved the equations expressing the height of the half-cube a as a function of R, in the two cases of (1) the area of the half-cube being equal to the area of the half-sphere or (2) the volume of the half-cube being equal to the volume of the half-sphere. These two situations give completely different results but with some sort of "connection" and here they are:the next step is to calculate in the first case (1) the volume of the half-cube and compare it with the volume of the half-sphere and in the second case (2) the area of the half-cube and compare it with the area of the half-sphere. These final steps have been by far the trickiest so that I had to keep checking the results over and over... before getting to something that resembled the expected outcome. I did both comparisons in the form of percentage proportion so in the first case of (1) the half-cube with the same area of the half-sphere, the volume of the half-cube is about 72.36% of the volume of the sphere. Here's the calculations.
In the second case of (2) the half-cube with the same volume of the half-sphere, the area of the half-cube is about 124.07% of the area of the half-sphere and here's the other calculations.
So at the very end the half-sphere (dome) wins on efficiency in both cases, encasing the biggest volume with a given area and the smallest area with a given volume! It all goes along with the rules of the Nature as they appear to be: we can all sleep well trusting that things are good, in the perfect world of theory...