True airspeed is how fast an aircraft moves through the surrounding air, and in thin air at altitude it is faster than the airspeed dial shows. The FAA's training handbook offers a shortcut: add 2% to calibrated airspeed for every 1,000 ft. On a standard day that shortcut changes sign near 25,900 ft, by our calculation. Below that height it gives too much, at most 3.24% at about 12,500 ft. At 36,000 ft it gives 6.1% too little. The true rise is about 1.5% per 1,000 ft near sea level and just under 2% at the tropopause, so a compounding curve is being matched with a fixed step.
Corrected in stages, the dial reading becomes true airspeed
The US definitions in 14 CFR 1.1 form a ladder. Indicated airspeed is the instrument's reading, calibrated to sea-level air. Calibrated airspeed (CAS) is that reading "corrected for position and instrument error". Equivalent airspeed (EAS) is CAS "corrected for adiabatic compressible flow for the particular altitude", and true airspeed (TAS) "is equal to equivalent airspeed multiplied by (ρ0/ρ)^1/2", the square root of sea-level density over actual density.
The FAA's Pilot's Handbook of Aeronautical Knowledge, chapter 8, skips the middle rung and calls TAS "CAS corrected for altitude and nonstandard temperature". Everything below is arithmetic about air; it describes no aircraft's performance, and the flight manual governs anything operational.
Defined at sea level and 15 °C, the standard atmosphere cools steadily to the tropopause
Table 4-2 of the FAA's Aviation Weather Handbook (FAA-H-8083-28, 2022), captioned "Selected Properties of the Standard Atmosphere", sets sea level at 1013.25 hPa (29.92 inHg) and 15 °C. Temperature falls 6.5 °C per 1,000 m, or 1.98 °C per 1,000 ft on our conversion, until the tropopause at 11,000 m (36,089 ft), where it reaches −56.5 °C. ICAO publishes its own standard atmosphere as Doc 7488, which we did not open.
Density altitude, in the same handbook, is "the pressure altitude corrected for temperature deviations from the standard atmosphere". The density ratio σ, actual over sea-level density, is what turns EAS into TAS.
Measured on a standard day, the FAA's shortcut runs high below about 26,000 ft
We computed TAS for 100 kt calibrated from the density ratio alone, in 10-ft steps, in dry air at standard temperature. Compressibility, a small extra at this speed, comes later.
| Pressure altitude | TAS from density ratio (kt) | Add 2% per 1,000 ft (kt) | Rule's error |
|---|---|---|---|
| 2,000 ft | 103.0 | 104.0 | +1.0% |
| 8,000 ft | 112.8 | 116.0 | +2.8% |
| 12,480 ft | 121.0 | 125.0 | +3.2% |
| 20,000 ft | 137.0 | 140.0 | +2.2% |
| 30,000 ft | 163.5 | 160.0 | −2.1% |
| 36,000 ft | 183.2 | 172.0 | −6.1% |
The overstatement peaks at 12,480 ft and reaches zero at about 25,900 ft. Above that the rule trails, by more with each thousand feet. None of the ranking pages we read tabulates the error by altitude and temperature.
Warmed well above standard, mid-level air makes the rule nearly exact
Warm air is thinner, so TAS rises with heat. At 8,000 ft, for instance, an extra 10 °C adds 2.05 kt to a calibrated 100 kt. Applied to pressure altitude, the rule matches air about 13–17 °C warmer than standard between roughly 6,000 and 15,000 ft, and at 15 °C above standard it lands within 0.5% from 8,000 to 15,000 ft. Across all heights to 36,000 ft, from 20 °C below standard to 30 °C above, its error ran from −12.0% to +7.4%.
Near sea level, where the rule adds nothing, heat works against it. Meteorological Service Singapore's climate page gives 32.4 °C as April's mean daily maximum at the Changi Climate Station over 1991–2020. Assuming standard pressure and dry air, σ is 0.9431 and 100 kt calibrated is 103.0 kt true, so the rule's 100 is 2.9% short. Our Changi density altitude guide puts that air at 2,017 ft dry, at the airport's 6.66 m elevation, and 2,417 ft once water vapour is counted; at the humid figure, 100 kt calibrated comes to about 103.6 kt true on our arithmetic. The Aviation Weather Handbook gives the reason: air with more water vapour "is less dense".
Applied to density altitude, the rule overstates between sea level and the crossover
The handbook's rule names no kind of altitude, though its flight-computer method takes "CAS, pressure altitude, and temperature". AVweb's 2001 Desired Track feature adds the 2% "to the indicated airspeed" and uses temperature "only to determine true altitude". A Boldmethod article by Colin Cutler, dated 28 May 2026, says true airspeed "increases about 2% per thousand feet of density altitude".
The choice moves the bias. Fed density altitude, the rule follows the standard-day error curve whatever the temperature, so it reads high at every density altitude between sea level and the crossover: +0.86% at sea level and +3.09% at 8,000 ft, both in air 15 °C above standard. Fed pressure altitude in that warm sea-level air, it reads 2.51% low. For the dry Changi afternoon, the density-altitude version gives about 104 kt, roughly 1% high.
Pushed to jet speeds at altitude, density arithmetic needs the compressibility step
Dividing CAS by the square root of σ treats it as EAS. The fuller route rests on NASA Glenn's isentropic flow equations: calibrated airspeed fixes an impact pressure, impact pressure over static pressure fixes the Mach number, and Mach times the local speed of sound is TAS. Where shocks form, NASA notes, "the isentropic relations are no longer valid".
At sea level, where EAS equals CAS, skipping it costs nothing; aloft the cost grows with speed. On a standard day the density-only answer runs 1% high from about 288 kt calibrated at 10,000 ft, and from only 124 kt at 30,000 ft. At 250 kt and 36,000 ft it gives 457.9 kt against 434.3 kt compressible (Mach 0.757), 5.4% high. Below 150 kt and 15,000 ft it stays under 0.5%. On our figures, the compressible answer at 100 kt moves the rule's crossover up to about 27,000 ft.
Until 21 October 2026 our true airspeed calculator used the density ratio alone with no Mach limit, so 450 kt calibrated at 36,000 ft on a standard day returned 824 kt, faster than sound at that height. The revision made that day converts through impact pressure and Mach, refuses inputs that reach Mach 1, and shows the density-only figure and the 2% rule beside its result.
Worked by hand, the density ratio needs only a square root
Take 100 kt calibrated at 8,000 ft pressure altitude and −1 °C outside, within a degree of standard. The pressure ratio is (1 − 0.00000687535 × 8,000) raised to the power 5.2559, or 0.7428. The temperature ratio is 272.15 K over 288.15 K, or 0.9445. This gives a σ of 0.7865, with a square root of 0.8868.
Dividing 100 by 0.8868 gives a density-only TAS of 112.8 kt. The 2% rule gives 116.0, 2.9% more. Carried through impact pressure and Mach 0.175, the compressible conversion gives 112.65 kt, a tenth of a knot lower. At this speed, the rule's error is about thirty times larger than the compressibility correction it ignores.
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