Pressure Is Set by Depth, Not Shape
Here are containers of every shape: a tall narrow one, one that flares wide at the top, one that bulges at the bottom. Switch between them with the buttons, and as long as you fill the water to the same height, the pressure gauge at the bottom reads the same value, even though the amount of water is clearly different. What sets the pressure is not how much water there is, but how deep below the surface the point sits. This is called the hydrostatic paradox. It looks strange, but it follows naturally from the last lesson, that pressure at a point has no direction. Now drag the depth down and you will see the pressure grow in direct proportion to it.
Switch the container shape one by one. The one that flares wide holds far more water, the narrow one far less. Yet the bottom gauge reads the same for all three. How? Pressure at the bottom does not feel only the column of water directly above it; at a given depth it receives push from the sides too, equally in every direction. In a wide container, the weight of the water spread out to the sides is carried by the slanted walls instead. So all that is left at the bottom is the pressure that depth alone sets. Neither the amount of water nor the width of the container matters.
Now grab a point and drag the depth h downward. The deeper it goes, the more the pressure grows, in a straight line. Go twice as deep and the pressure is exactly twice as large, no curve, just a straight proportion. Just below the surface it is almost zero; as you descend it adds up at a steady rate. This straight, proportional relationship is the heart of hydrostatic pressure: it means you never need a table, just the depth, to read off the pressure.
Why depth, and why exactly in proportion? Think about the weight of the column of water sitting above the point. A column of cross-section A and height h has volume A times h, and weight equal to density times gravity times volume, that is ρ g A h. This weight presses on the bottom area A, so the pressure is weight divided by area. But the A on top and the A on the bottom cancel. What remains is p = ρ g h. Because the area dropped out, it makes no difference whether the container is fat or thin. Drag h and watch the weight and the pressure grow together as the column gets taller.
The formula has one more ingredient besides depth: the density ρ. At the same depth, a heavier liquid builds pressure faster. Drag the density up from oil to water to seawater and watch the slope of pressure-versus-depth get steeper. That slope is exactly ρ g. Seawater presses a little harder than fresh water, and mercury presses 13.6 times as hard as water at the same depth. That is why old barometers used mercury rather than water: a water column would need to be 10 meters tall, while mercury needs only 76 centimeters.
Now go into the sea. As a diver descends, the depth h grows, and the water pressure on the body grows as ρ g h. A handy rule of thumb: for every 10 meters of water, the pressure adds about one atmosphere. Since the air already presses with one atmosphere at the surface, at 10 meters the total is about two atmospheres, and at 20 meters about three. That is why your ears feel it. It is also why a dam gets thicker toward its base, and why a submarine has a depth limit, all because of this pressure from depth. Drag h to send the diver down and watch how the pressure stacks up.