In a Still Fluid, Pressure Is the Same in Every Direction

First, get a feel for the pressure gathered at one point. Each arrow is the pressure pushing on that point. Drag θ and the highlighted direction sweeps around. Yet however you turn it, every arrow stays the same length. Length is the size of the pressure, and changing direction does not change it. Why? A fluid at rest cannot push harder in just one direction. If one direction were stronger, the fluid would have flowed that way. Sitting still, not flowing, means every direction is in exact balance.

Let us prove that balance with a single tiny wedge. We have cut a very small triangular sliver out of the fluid. On each of its three faces, pressure presses straight in: p_x on the vertical face, p_y on the bottom face, p_n on the slanted face. Drag θ to tilt the wedge. The areas of the faces all change, yet the three pressures stay equal. The secret is force balance. The force on the slanted face is pressure times area, and its horizontal part matches the vertical-face force while its vertical part matches the bottom-face force. Where geometry enlarges an area, the angle trims it back by just as much, so in the end p_x = p_y = p_n. This is the root of Pascal's principle.

So what exactly is pressure? It is the force pressing perpendicular on a face, divided by the area of that face. p = FA. The form is the same as stress, but fluid pressure has two special traits. First, it always pushes perpendicular to the face. Second, it always pushes, never pulls. Drag A. The force F stays fixed, but as the face widens the pressure drops and turns blue, and as it narrows the pressure climbs and turns red. The same force funneled onto a small face makes the pressure shoot up. A thumbtack bites because of this division, and a wide ski keeps you on top of the snow for the same reason.

Now we tie the earlier pieces together. Place a small plane through the point and rotate it with φ. At whatever angle you turn it, the pressure p on that plane reads exactly the same. If turning the face does not change the reading, then pressure has no direction. A force is a vector with a direction, but pressure is a scalar with none. That is why we write pressure as a single number, not an arrow. Only once you name a face does the number become a force: multiply it by the area and attach the face's perpendicular direction. Pressure itself is just one state of that point.

This one fact hides all around us. Pick the balloon. The air inside pushes on every point of the skin equally and perpendicular, so the balloon swells into a round shape. The submarine? Deep water presses perpendicular and equal on every part of a rounded hull, so a circular cross-section resists pressure best. Submerge a flat box and it is the same. Even where the wall is flat, at each point of that wall the pressure acts perpendicular to the face. Whatever the shape, a fluid pushes perpendicular on every face it touches, and at a given point with equal size in every direction. This is the ground that the next stories, pressure with depth and buoyancy, are built on.