Hydrostatic Equation Pressure Calculator
Find atmospheric pressure at a new height, using either the simple linear hydrostatic equation or the full barometric exponential formula.
📉 What is the Hydrostatic Equation?
The hydrostatic equation is dP/dz = -ρg, the fundamental statement that atmospheric pressure decreases with height at a rate set by local air density and gravitational acceleration. It comes from balancing the downward weight of the air above any point against the upward push of the pressure difference holding that air up, in a fluid at rest, those two forces are always in balance.
Atmospheric scientists and pilots use the hydrostatic relationship constantly, as the basis for pressure altitude and density altitude calculations used in every flight, as the foundation for converting surface weather-station pressure readings to a common sea-level reference for weather maps, as the starting point for deriving the barometric formula used in engineering and geophysics, and as the physical reason a mountain climber's ears pop and an unpressurized aircraft cabin needs supplemental oxygen at altitude.
A common misconception is that pressure decreases with height in a straight line all the way up through the atmosphere. It does not, the simple linear relationship (constant density) is only a good approximation over a few hundred meters, over larger height ranges the barometric exponential formula is needed, since air density itself thins out as you go up, which in turn changes how fast pressure keeps dropping.
This calculator provides both forms: a simple linear mode for quick estimates over modest height changes, and the full barometric exponential formula for larger height changes where the changing density can no longer be ignored.
📐 Formula
Both forms derive from the same hydrostatic balance (Wikipedia, "Vertical pressure variation"); the barometric formula additionally applies the ideal gas law to let density vary with height, per the standard barometric formula reference.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Simple Mode, Standard Ascent
P₀ = 1013.25 hPa, ρ = 1.225 kg/m³, Δh = 500 m
Example 2 — Simple Mode, Larger Ascent
P₀ = 1013.25 hPa, ρ = 1.225 kg/m³, Δh = 1500 m
Example 3 — Simple Mode, Lower Base Pressure
P₀ = 1000 hPa, ρ = 1.1 kg/m³, Δh = 300 m
Example 4 — Barometric Mode, Standard Day at 1000 m
P₀ = 1013.25 hPa, h = 1000 m, T = 15°C
Example 5 — Barometric Mode, Higher Altitude
P₀ = 1013.25 hPa, h = 3000 m, T = 15°C
Example 6 — Barometric Mode, Cold Day
P₀ = 1013.25 hPa, h = 1000 m, T = -10°C
❓ Frequently Asked Questions
🔗 Related Calculators
What is the hydrostatic equation?
The hydrostatic equation is dP/dz = -rho g, it states that atmospheric pressure decreases with height at a rate equal to local air density times gravitational acceleration. It is the fundamental balance between the downward pull of gravity and the upward push of the pressure gradient in a fluid at rest.
What is the difference between the simple and barometric modes?
The simple (linear) mode assumes air density stays constant over the height change, giving deltaP = -rho g deltah, accurate for small height changes. The barometric (exponential) mode, P = P0 exp(-gMh / RT), accounts for the fact that density itself decreases with altitude, making it accurate over much larger height ranges.
What is the barometric formula?
The barometric formula is P = P0 exp(-gMh / (RT)), where P0 is the base pressure, g is gravity, M is the molar mass of dry air, h is height above the reference level, R is the universal gas constant, and T is absolute temperature in kelvin. It derives from combining the hydrostatic equation with the ideal gas law.
Why does the barometric formula need a temperature input but the simple mode does not?
The barometric formula derives pressure from the ideal gas law, which links pressure, density, and temperature together, so temperature is required to compute how density (and therefore the pressure decay rate) changes with height. The simple mode sidesteps this by taking a fixed density value directly instead of deriving it from temperature.
How accurate is the simple linear mode compared to the barometric mode?
For height changes of a few hundred meters, the simple linear mode and the barometric mode agree closely, since air density barely changes over that range. Beyond a kilometer or so, the simple mode increasingly overestimates the pressure drop, because it holds density fixed at the base-level value instead of letting it decrease with altitude like the barometric formula does.
What values does this calculator use for g, M, and R?
g = 9.80665 m/s2 (standard gravity), M = 0.0289644 kg/mol (molar mass of dry air), and R = 8.31446 J/(mol K) (universal gas constant), the standard values used throughout the International Standard Atmosphere model.
Can the height change be negative in the simple mode?
Yes. A negative height change (descending) produces a positive pressure change, since pressure increases as you go down. For example, descending 500 m below the reference level increases pressure by the same magnitude that ascending 500 m would decrease it.
Why is there no maximum height limit on the barometric mode?
The formula itself has no built-in limit, but its isothermal assumption (constant temperature through the layer) becomes less realistic over very large height ranges, since real atmospheric temperature changes with altitude. It is most accurate within a single roughly isothermal layer, such as the lower troposphere or the tropopause region.
How does pressure change with a 1000 m increase in height near sea level?
Under standard conditions (1013.25 hPa, 15 degrees C), pressure drops from 1013.25 hPa to roughly 900 hPa over the first 1000 m, about an 11 percent decrease, a useful rule of thumb for the lower atmosphere.
Is this the same formula used for pressure altitude and density altitude?
It is the same underlying hydrostatic physics, but used in the opposite direction. The Pressure Altitude and Density Altitude calculators on this site start from a known pressure or altimeter setting and solve for altitude, this calculator starts from a known altitude (or height change) and solves for pressure.
What units does this calculator use for pressure?
Pressure is entered and displayed in hectopascals (hPa), numerically identical to millibars (mb) and the standard unit used on weather maps and by most aviation altimeters outside North America.