hydrostatic pressure vs depth
Here are vessels of different shapes. A narrow tall one, one that flares at the top, one that bulges at the bottom, and when you switch the vessel with a button and fill the water to the same height, the pressure gauge at the bottom points to the same value. The amount of water in the vessel is clearly different, and still it is so. What sets the pressure is not how much water there is, but how deep that spot is below the surface, and this is called the hydrostatic paradox. It follows naturally from the fact that the pressure at one point does not depend on direction. Drag the depth down and you see the pressure grow straight with depth. A water surface at the same height presses the bottom with the same strength. The silhouette of the vessel never enters that number.
Switch the vessel shapes one by one and the flared vessel holds much more water and the narrow vessel holds less, while the bottom gauges of all three read the same. How? The pressure at the bottom does not feel only the water column sitting right above that point. At the same depth it also receives, equally from every side, the push coming from the side. The weight of water that spread sideways in a wide vessel is taken instead by the slanted wall, so what remains at the bottom is only the pressure that depth sets. Neither the amount of water nor the width of the vessel matters. The wall takes the sideways weight and the bottom takes only the depth. That is why the gauge needles gather in one place.
Now take one point and drag the depth h down, and the deeper you go the more the pressure grows straight. Go twice as deep and the pressure is exactly twice, and it does so in a straight line, without bending. Just under the surface it is nearly 0, and as you go down it adds at a steady rate, and this straight proportion is the heart of hydrostatic pressure. If you know only the depth, you can read the pressure at once. A straight line means that when you read the depth the pressure follows. One depth is enough, with no table.
Why is it proportional to depth of all things? Think of the weight of the water column sitting on that point and the answer comes. A water column of cross section A and height h has volume A times h. The weight is density times gravitational acceleration times volume, that is ρ g A h. This weight pushes the bottom area A, so the pressure is weight divided by area, and A cancels top and bottom and what remains is p = ρ g h. The area vanished, so it does not matter whether the vessel is thick or thin, and when you drag h you see the weight and the pressure grow together as the column grows. After the thickness cancels, only height and density remain. A longer column means a longer push.
The formula has one more thing besides depth, the density. Even at the same depth a heavier liquid grows its pressure faster, and when you drag the density up from oil to water to seawater the slope at which pressure rises with depth gets steeper. That slope is exactly ρ g. Seawater pushes a little harder than fresh water, and mercury pushes 13.6 times water at the same depth, and that is why old barometers used mercury rather than water. A water column would need 10 meters, while mercury is enough at 76 centimeters. A heavier liquid makes a steeper slope at the same depth. That is why mercury holds with a short column.
Now go into the sea. As a diver goes down the depth h grows and the water pressure on the body grows as ρ g h, and a handy estimate is that about every 10 meters down in water the pressure adds 1 atmosphere. At the surface the air already pushes 1 atmosphere, so at 10 meters the sum is 2 atmospheres, and at 20 meters it is 3 atmospheres. This is why the ears go muffled, why a dam grows thicker toward the bottom, and why a submersible sets a depth limit. Drag h to send the diver down and you see how the pressure stacks. The farther down you go, the harder the water column holds the body. The thickness of a dam and the muffling of an ear point along the same line.