Dome vs cube: the math of the results

In this post I would like to share the results of my mathematical meditations on the geometry of domes and cubes, considering the beauty of perfect conditions that all the proper rules give. It's all aimed at studying geodesic domes as an approximation of the half sphere.
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...