A Mathematical Explanation of Buoyancy

Let's consider a box with length=L, width=W, depth=D. We can compute the net buoyant force by adding up the contributions from the pressure acting on all the sides of the box. Remember that force = pressure x area. The force on the left and right sides cancels out, and the force on the front and the back cancels out. This leaves just the force on the top and the bottom. The pressure varies with depth, of course, so the pressure on the bottom is greater by an amount equal to the fluid density x gravitational acceleration x change in depth.

We can compute the buoyancy with the same box oriented horizontal or vertical and the buoyant force will come out the same! Buoyancy depends on size but not on shape.

Buoyancy