That would be volume, not area. certainly not "surface" area.
Even if you fill a ball with air, it will weigh more than an empty ball, but then why does the scale ever read zero if there is allways air?
Even a floating ball will add to the total weight of the tank, by the weight of the ball & the air.
Now; here is a stumper question for you.....
If you suspend your (10 lb) arm in the water without touching the tank (only water), will that add to the weight of the tank? If so, how much? If not, then when you grab the 20 pound fish in your hand does the weight go back to 100 lbs? (even with your arm still inside) hmmmm....
Come on CHOMPERS this one is for you

just cause I like you.