Air (or water) pressure cannot possibly cause what we call "weight" simply because air is a fluid.
Pressure in a fluid always causes a force at right-angles to the surfaces of any objects in it.
So if we have a horizontal flat sheet the force pressing down on the top surface is balanced by an equal force pressing up on the bottom.
It does not matter how high or low the pressure is the same thing applies, as long as the pressure on the top is the same as the pressure on the bottom. So the pressure does not cause any nett downward force, so does not contribute to weight.
Now we know from measurements that the air pressure gets lower as we increase in altitude.
Intuitively it would seem that this cannot be a very great amount, but it is more than we realise. A helium balloon floats simply because the air pressure pressing up from underneath is slightly greater than the air pressure pushing it down.
The amount of this lift can easily be calculated. A cubic metre of air (at STP
[1]) weighs 1.225 kg, so the air pressure decreases by 1.225 kg per square metre for each metre increase in altitude. This sounds a lot until it is realised that the normal air pressure is 10,340 kg/m
2.
So back to our funny cube shaped helium balloon, the total force caused the pressure on the bottom is 1.225 kg more than the force on the top, so there is a nett uplift of 1.225 kg.
Now of course we have to subtract from this the weight of the balloon. If we can neglect the weight of the balloon skin, the weight of the helium would be its density (0.164 kg/m
3) times its volume (1 m
3), or 0.164 kg.
So a 1 cubic metre helium balloon has a buoyancy of just over 1 kg, less the weight of the balloon structure.
In the end the shape of the balloon makes no difference and the "lift" is simply the
(density of the air displaced) x (the volume of the object)
In other words Archimedes Principle.
So, certainly density comes into working out buoyancy,
but pressure cannot cause weight, because the forces caused by pressure act equally in all directions.
[1] STP is short for
Standard
Pressure and
Temperature, ie sea level, etc.