Hong Life CalculatorsUnderstanding the barometric altitude and ISA atmosphere calculator — concept and principle
This calculator gives the pressure, temperature and air density of the International Standard Atmosphere (ISA) at a chosen altitude, and can also work backwards from a measured pressure to a "pressure altitude". The ISA is the reference used to calibrate altimeters and compare aircraft performance.
How it is calculated
For the troposphere (0–11 km), with altitude h in metres:
- Pressure p = 1013.25 × (1 − 2.25577 × 10⁻⁵ × h)^5.25588 hPa
- Temperature T = 15 − 0.0065 × h °C
- Density ρ = 1.225 × (1 − 2.25577 × 10⁻⁵ × h)^4.25588 kg/m³
- Altitude from pressure h = 44,330.8 × (1 − (p / 1013.25)^0.190263) m
The principle
The ISA assumes dry air as an ideal gas, sea-level values of 15 °C, 1013.25 hPa and 1.225 kg/m³, and a temperature that falls linearly at 6.5 °C per kilometre (the lapse rate). Each thin layer of air must support the weight of the air above it (hydrostatic balance, dp = −ρ g dh). Integrating this with the ideal gas law and the linear temperature profile gives a power law. The constant 2.25577 × 10⁻⁵ is 0.0065 ÷ 288.15 K, and the exponent 5.25588 is g M ÷ (R L). Density falls more slowly than pressure because the air also gets colder, hence the exponent is one smaller.
Worked example
- Input: h = 3,000 m; measured pressure 0 (skipped).
- Base: 1 − 2.25577 × 10⁻⁵ × 3,000 = 0.9323269.
- Pressure: 1013.25 × 0.9323269^5.25588 = 1013.25 × 0.69192 = 701.0852041 hPa, about 69% of sea level.
- Temperature: 15 − 0.0065 × 3,000 = −4.5 °C.
- Density: 1.225 × 0.9323269^4.25588 = 0.9091217816 kg/m³, about 74% of sea level.
- The pressure altitude shows "—" because no measured pressure was entered.
| Altitude (m) | ISA pressure (hPa) |
|---|---|
| 750 | 926.3360365 |
| 1,500 | 845.5599052 |
| 2,250 | 770.584357 |
| 3,000 | 701.0852041 |
| 3,750 | 636.7510339 |
| 4,500 | 577.2829233 |
| 6,000 | 471.8099341 |
How to read the result
Near sea level, pressure falls about 12 hPa for every 100 m of climb. Roughly half of the atmosphere's mass lies below 5.5 km, where ISA pressure is about 505 hPa; at the summit height of Everest (8,849 m) it is about 314 hPa. The oxygen fraction stays near 21%, but each breath contains fewer molecules, which is why altitude sickness becomes common above about 2,500 m.
Common mistakes
- Treating the result as the actual weather; real pressure and temperature vary daily by tens of hectopascals and many degrees.
- Using the formulas above 11 km, where the ISA temperature becomes constant and a different equation applies.
- Using station pressure and sea-level pressure interchangeably when estimating altitude.
Frequently asked questions
What is pressure altitude?
The altitude in the ISA at which the measured pressure would occur. Aircraft fly flight levels based on it, using 1013.25 hPa as the reference.
Why does the boiling point of water drop on mountains?
Water boils when its vapour pressure equals the surrounding pressure, so lower pressure means a lower boiling point; at 701 hPa it is roughly 90 °C.
Sources: ICAO Doc 7488/3, Manual of the ICAO Standard Atmosphere, 1993 · ISO 2533:1975, Standard Atmosphere · U.S. Standard Atmosphere, 1976 (NOAA, NASA, USAF)