@Jackblack
In a side surface of 2Χ2, after 3 times of 0.7928932 roundings there comes 1.9939033 which is a square on rounded sphere. And if you translate it into flat round area, you got r=0.7928932.
You need to match it with rounded dome by substracting its area. The round radius is still (0.7928932), but the base radius got changed from 0.7928932 to be 0.7071068, so the calculation will be dome area = flat round area Χ (0.7071068/0.7928932) = 1.77817