This page assumes nothing. Before you can read the parent derivation, you must be fluent with eight small ideas. We build them one at a time, each on top of the last, each anchored to a picture. If you already know some, skim — but every symbol the parent uses lives here.
Why the topic needs it. A fluid presses on the walls and floor of its container. That press is a force. If we cannot talk about pushes and pulls, we cannot say a single thing about pressure.
One special force appears constantly: weight. Weight is the downward pull of gravity on an object's mass. We will build it properly in step 5 once we have mass and gravity.
Why the topic needs it. The same push spread over a big surface feels gentle; the same push concentrated on a tiny surface feels sharp. Pressure is exactly this "push per surface," so we must be able to measure the surface. In the derivation, the top and bottom of the fluid column each have area A.
Look at the amber arrow (the force) and the cyan patch (the area). Same arrow, two patch sizes — the small patch gets crushed, the big patch barely notices. That is the whole meaning of "per area."
We chose force ÷ area (not force ÷ something else) because the thing we actually feel underwater is not the total push on the whole pool floor — it is how concentrated the push is on our own small patch of skin.
We will reverse this relation later: if we know a pressure P acting on a face of area A, the force on that face is F=P×A. In the parent, this is used to turn each pressure into a force on the column's top and bottom faces.
Three ideas travel together. We need them because pressure comes from weight, and weight needs mass, and the mass of a fluid comes from its density times its volume.
Why the topic needs it. In the derivation we cut out an imaginary column of fluid. To find its weight we first need its mass, and with constant density the mass of that column is simply m=ρV=ρ(Ah).
Why the topic needs it. Pressure with depth exists only because the fluid is heavy. Combining our tools — W=mg, then m=ρV, then V=Ah — the weight of a fluid column is
W=mg=(ρV)g=ρ(Ah)g,
which is the heart of the whole derivation.
The figure shows the three vertical arrows on the column. Notice the two sideways cyan arrows on the wall: they are horizontal, so they cancel left-against-right and never enter the vertical balance. That is why only the top and bottom faces matter.
Read it upward: force and area make pressure; area and depth make volume, which with constant density makes mass, which with gravity makes weight; equilibrium ties pressure and weight together into the final formula.
