An A380 has up to 2 decks for passengers and only 1 deck for cargo (for passenger's luggage), whereas a B777 has 1 deck for passengers and 1 deck for luggages. It looks like the volume of luggage per passenger is reduced in a A380 compared to a B777. Is that true? Is the volume of luggage per passenger the same? Is there a method to calculate the volume of the cargo and then to design its shape?
The cargo hold volume is a secondary result of the design process. Pressurized fuselages need to be round for best structural efficiency, and the passenger compartment needs to be high enough for passengers to stand upright. The rest has to follow from that, and the old narrow-body aircraft (like the DC-8 or the Boeing 707) were mostly volume-limited (which means that they would have benefited from more internal volume), whereas the early wide-bodied aircraft (A300, Boeing 747) are weight-limited (which means operators flew them with some volume unfilled to avoid overloading the plane).
The picture shows the fuselage cross section of an A310 (left) and a Boeing 737 (right), which are typical for wide-bodied rsp. narrow-bodied designs. Note the relatively bigger freight compartment of the Airbus fuselage.
Even bigger fuselages than those of the Boeing 777 or the MD-11 offer so much internal volume that two passenger decks are needed to avoid flying lots of empty space around. McDonnell-Douglas planned for a while to expand the passenger load of the MD-11 by adding a lower passenger deck forward of the wing, and Airbus did the logical thing and designed the A380 from the outset as a double-decker. Other designs of a similar size came to the same conclusion.
In all cases the cargo volume was determined by what was left unfilled.
The cargo compartments are not only for luggage. Commercial cargo also travel on passenger aircraft, besides cargo aircraft. With luggage alone, there are plenty of spaces in the cargo deck.
In fact, if you fill all the seats, all the cargo space and sufficient fuel for a long trip in a B777, the plane won't takeoff - its GW (Gross Weight) exceeds the MTOW (Max Take-Off Weight).
Airlines are smart people - if a flight is not full, dispatch will put more cargo on the aircraft to utilize it. A lot of passenger airliners also offer air delivery service.
Some passengers complain, "I can't book the flight. The staff says it's full, yet there are many empty seats!" Commercial cargo yields more revenue than an economy seat. Thus, given the weight limitation, an airline may prefer cargo vs passenger.
I don't have info about A380, but I'd guess that the A380 has less extra cargo space than the B777. But as long as it's enough, it isn't a problem.
Cargo compartments in airliners are generally in two categories: regular cargo and bulk cargo. This graphic shows the compartments in a 777.
The normal cargo compartments are generally filled with containers called Unit Load Devices. These are standard containers that can be filled with cargo and easily loaded/unloaded from the cargo compartment. Palletized cargo can also be loaded.
The bulk cargo compartment is for non-containerized cargo like loose luggage. You can see that the majority of cargo space is reserved for non-bulk cargo.
In the case of the A380, there is a large bulk hold as well as plenty of room for other cargo.