Haesslich: I got the 290 megatons (vs. 290 kilotons) figure based on the idea that 290 kilotons seems an appropriate mass for the dreadnought if it were the 2.2 km size, and scaling that up by a factor of ten in all dimensions would mean one thousand times the mass.
As for calculations, let's say for the sake of argument that the dreadnought's hull actually takes up about ten percent of the volume of the box that circumscribes it (about 22 km by 5 km by 5 km if we go with the larger size). If anybody has a better guess than ten percent and a reason for it, then please correct me.
Anyway, this gives us an estimated hull volume of about 50 cubic kilometers. Let's cut this even further by assuming that 50% of the space is the hangar/landing bay(s), which are depressurized and hence do not contribute to the active volume. This leaves us with 25 cubic kilometers.
Earth sea-level air at human-comfortable temperature masses 1.2 kilograms per cubic meter. 25 cubic kilometers of it would thus mass approximately sixty megatons. Even by cutting the air mass to a minimum by making it pure oxygen would still require at least a fifth of that mass. Note also that even though not all of the space inside the dreadnought is air (there are bulkheads and equipment and other things), pretty much everything except for vacuum spaces that would be inside the dreadnought would be DENSER than air.
This leads me to the conclusion that either the dreadnought masses AT LEAST tens of megatons or else it is largely a hollow shell. This is why I believe 290 megatons to be a reasonable figure.