Air is a gas, but for purposes of calculating atmospheric pressure, you could regard it as a fluid, and calculate the pressure at sea level using the expression for fluid pressure. This expression is P = ∂gh, where ∂ is the density of air, g is the acceleration of gravity, and h is the height of the atmosphere. This approach doesn't work, though, because neither ∂ nor h are constant. The traditional approach is to measure the height of a column of mercury instead. If you're looking for atmospheric pressure at a particular altitude, you could use the barometric formula. This is a fairly complex relationship that depends on several variables, so it's easier to just look up the value you need in a table.
TL;DR (Too Long; Didn't Read)
Scientists calculate the atmospheric pressure at sea level by measuring the height of a column of mercury and calculating the pressure the atmosphere has to exert to raise the column to that height.
The Mercury Barometer
Immerse a glass tube with a closed end in a tray of mercury and allow all the air to escape, then turn the tube upright with the opening submerged in the mercury. You'll have a column of mercury inside the tube and a vacuum between the top of the column and the end of the tube. The pressure exerted by the atmosphere on the mercury in the tray is supporting the column, so the height of the column is a way to measure atmospheric pressure. If the tube is graduated in millimeters, the height of the column will be approximately 760 mm, depending on atmospheric conditions. This is the definition of 1 atmosphere of pressure.
Mercury is a fluid, so you can calculate the pressure needed to support the column by using the equation P = ∂gh. In this equation, ∂ is the density of mercury and h is the height of the column. In SI (metric) units, one atmosphere is equal to 101,325 Pa (Pascals), and in British units, it's equal to 14.696 psi (pounds per square inch). The torr is another unit of atmospheric pressure originally defined to be equal to 1 mm Hg. Its current definition is 1 torr = 133.32 Pa. One atmosphere = 760 torr.
The Barometric Formula
Although you can't derive atmospheric pressure at sea level from the total height of the atmosphere, you can calculate changes in air pressure from one height to another. This fact, along with other considerations, including the ideal gas law, lead to an exponential relationship between the sea level pressure (P0) and pressure at height h (Ph). This relationship, known as the barometric formula, is:
Ph = P0e-mgh/kT
- m = mass of one air molecule
- g = acceleration due to gravity
- k = Boltzmann's constant (ideal gas constant divided by Avogadro's number)
- T = temperature
Although this equation predicts pressures at various heights, its predictions differ from observation. For example, it predicts a pressure of 25 torr at a height of 30 km (19 mi), but the observed pressure at that height is only 9.5 torr. The discrepancy is primarily due to the fact that temperatures are colder at higher elevations.