Atmospheric Pressure Calculator

What Is the Atmospheric Pressure Calculator?

How to use the atmospheric pressure calculator

1. Pick the unknown. The Solve for menu offers Pressure at altitude (P), the default, or Altitude (h) for the altimeter problem. Whichever you choose, Sea-level pressure stays on the form as an input.
2. Enter the altitude. Type it in the Altitude box and pick m, km or ft from its drop-down; the tool accepts 0 to 20,000 m and returns nothing outside that band. When solving for altitude this box is replaced by Pressure at altitude, which takes kPa, hPa, Pa, atm, mmHg or psi.
3. Set the day's sea-level pressure. The default 1013.25 hPa is the standard atmosphere. For a real day, enter the mean sea-level pressure from a weather report, not the station reading, and choose hPa, kPa, Pa, atm, inHg or psi to match the source.
4. Read the result and the extras. The answer shows in the unit selected on the result field with the working underneath, followed by the ISA air temperature, air density, equivalent mercury column and the share of sea-level pressure at that height.

The common mistake is mixing up the two pressures: Pressure at altitude is what a barometer reads where you are standing, while Sea-level pressure is the value reduced to sea level that forecasts quote. Enter a station reading as P₀ and every altitude comes out too low.

Worked example: change one thing at a time

Start from the page defaults and move a single input per row. Every figure in the table is the string the calculator itself displays for those inputs, so if the table and the tool ever disagree, the tool is right.

Rows 1 to 4 climb through the troposphere and into the stratosphere with the standard 1013.25 hPa at sea level: the pressure drops to 78.46 % of sea level by 2000 m, under a third on Everest, and 5.40 % at 20 km. The temperature column falls 6.5 °C for every kilometre until 11 km and then freezes at -56.5 °C, which is why rows 3 and 4 share it. Row 5 keeps the altitude at 2000 m and lowers the sea-level pressure to 1000 hPa: the pressure, density and mercury column all scale down by the same 1.3 %, while the share of sea level and the temperature do not move at all.

Row 6 is the storm preset: at sea level the result is simply the 950 hPa you typed, so the tool reports 95 kPa and 100.00 %. Run the inverse to close the loop. Choose Altitude (h), enter 79.4952 kPa with the standard sea-level pressure, and the calculator returns 2000 m, the row 1 altitude recovered to the metre.

Formula and symbol reference

Everything the tool computes comes from the two-layer standard atmosphere: P = P0·(1 - L·h/T0)^(g·M/(R·L)) up to 11 km and P = P11·exp(-g·M·(h - 11000)/(R·T11)) above it, with the density from rho = P·M/(R·T). The constants are fixed; only P₀ and the altitude or pressure are yours to change.

The physics: why pressure falls with height

Air at rest obeys the hydrostatic balance dP/dh = -rho·g: cross a thin layer upwards and the pressure drops by the weight of that layer per unit area. For water the density is constant and this integrates to the familiar rho·g·h, but air is compressible, so its density is itself set by the pressure.

The ideal-gas law supplies the link, rho = P·M/(R·T), and substituting it gives dP/P = -(g·M/(R·T))·dh. If the temperature were constant the solution would be a pure exponential with a scale height of about 8.4 km at 15 °C.

The troposphere is not isothermal: the standard model cools it at the lapse rate L = 6.5 °C per km. Integrating with T = T₀ - L·h turns the exponential into the power law (1 - L·h/T0)^5.2559, and because the exponent is large the two shapes stay close, so the profile is nearly exponential.

At 11 km the model switches to a constant -56.5 °C and the exponential form takes over exactly, with the 11 km pressure of 22.63 kPa as its starting point. The calculator applies whichever branch the altitude falls in, and inverts the same branch when you ask for altitude from pressure.

Where the standard atmosphere breaks down

The formula describes an average dry atmosphere with a fixed temperature profile. Real air differs from it in a handful of predictable ways, and each one has a recognisable effect on the number you get.

Non-standard temperature

The lapse rate and the 15 °C sea-level temperature are averages. A hot summer column is less dense, so pressure falls more slowly with height and the true altitude for a given pressure is higher than the tool reports; a cold winter column does the opposite. The error is roughly 0.4 % of the altitude for every degree the column differs from standard.

Symptom: the pressure you measure at a known height sits a few per cent off the calculated value even with the correct P0 entered.

Weather systems and the sea-level pressure

The model has no weather; you supply it through P₀. Mean sea-level pressure ranges from about 950 hPa in a deep depression to 1050 hPa under a strong anticyclone, and every result scales with it. Use the day's MSLP from a report, and remember it drifts by a few hectopascals over a day, which is a few tens of metres of apparent altitude.

Humidity

The molar mass 0.0289644 kg/mol is for dry air. Water vapour is lighter than the nitrogen and oxygen it displaces, so humid air is slightly less dense and pressure falls slightly more slowly with height. The effect is usually below one per cent, up to about two per cent in hot saturated air, and the calculator ignores it.

Above 20 km

The isothermal layer ends at 20 km; above it the standard atmosphere warms again through the stratosphere with new lapse rates and new exponents. The calculator returns nothing rather than extrapolate, so an altitude above 20,000 m or a pressure below about 5.47 kPa gives the empty-result message.

Altimeter setting: QNH versus QFE

An aviation altimeter set to QNH reads altitude above mean sea level, which is what this tool computes when P₀ is the QNH. Set to QFE it reads height above the airfield, and set to the standard 1013.25 hPa it reads pressure altitude, the flight-level convention. Entering QFE as P₀ will make the calculated altitude a height above the field, not above the sea.

Geometric versus geopotential height

The standard atmosphere is tabulated against geopotential altitude, which treats g as constant at 9.80665 m/s². Real gravity weakens with height, so geometric height runs slightly larger: about 0.3 % more at 20 km, or 63 m. For a map altitude below a few kilometres the difference is smaller than the weather-driven error.

Where it is actually used

Aircraft altimeters and flight levels

A pressure altimeter is this calculator's altitude mode built into an instrument: it measures static pressure and displays the standard-atmosphere altitude for the P₀ set in its subscale. Above the transition altitude every aircraft sets 1013.25 hPa, so flight levels are pressure altitudes, and separation works because all aircraft share the same error.

Weather stations reducing to sea level

A barometer at a station 300 m up reads roughly 36 hPa less than the sea-level value, and the station adds that back using a formula of this kind so that maps compare like with like. Run the tool with the station height and read the share of sea level to see the size of the correction; operational reductions also fold in the measured temperature, which this tool does not.

Cabin pressurisation

Airliner cabins are typically pressurised to the equivalent of around 2,400 m rather than to sea level. Enter 2400 m and the tool reports the cabin pressure a passenger breathes at cruise, about three-quarters of sea level, while outside at 11,000 m the row 3 figure of 22.63 kPa applies. Check the specific aircraft type before quoting a cabin altitude, as the figure varies by design.

Cooking and boiling at altitude

Water boils where its vapour pressure equals the air pressure, so lower pressure means a cooler boil and longer cooking times. The tool gives you the pressure; the boiling point then comes from steam tables, roughly 93 °C at the 79.5 kPa of the 2,000 m worked example and near 70 °C at the Everest summit. Verify the pairing against a steam table before using it for anything that matters.

Barometric altimeters in phones and watches

Many phones, fitness watches and hiking GPS units carry a pressure sensor and convert it to altitude with exactly the inverse formula in the Altitude mode. They are precise to a metre or two over short times but drift with the weather, which is why serious units ask you to recalibrate P₀ at a known height, and why an app that fetches local MSLP from a weather service tracks better than one that assumes 1013.25 hPa.

Where to go next

For the concept itself, how a barometer works and why we do not feel the weight of the air, read the guide Atmospheric Pressure Explained and the broader pressure in physics article. To put numbers to force over area or to depth in a liquid, use the pressure calculator, and for the density of a gas at any pressure and temperature the gas density calculator. Watch pressure build with depth in the pressure simulator, or browse the library of physics simulations.