Why indicated airspeed is not your speed
An airspeed indicator is a differential pressure gauge. It compares the total pressure in the pitot tube with the static pressure at the static ports and displays the difference on a scale calibrated for sea-level standard density. That difference — impact pressure — is what the wing feels, so indicated airspeed is exactly the right number for flying the aeroplane: stall speed, best-glide speed, flap limits and manoeuvring speed are all fixed indicated values regardless of altitude, because they are all really statements about dynamic pressure.
What indicated airspeed is not is your speed through the air. At 20,000 ft the air has about half the sea-level density, so producing the same impact pressure takes roughly 40% more true speed. True airspeed is the number you need for navigation: it goes into the wind triangle, it sets your time en route, and combined with fuel flow it gives you range.
Between the two sit two more definitions worth knowing. Calibrated airspeed is indicated airspeed corrected for the position and instrument errors of your particular airframe; the correction table is in the flight manual and is usually a few knots, largest at low speed and high angle of attack. Equivalent airspeed is calibrated airspeed corrected for compressibility — the fact that air piling up in the pitot tube is squeezed rather than merely stopped. Below roughly 200 kt and 10,000 ft the compressibility correction is under a knot; at FL350 and 280 kt CAS it is more than 16 kt, which is why airliners are structurally limited on equivalent rather than calibrated airspeed.
The formula, in three moves
First, recover impact pressure from calibrated airspeed. By definition, calibrated airspeed is the speed that would produce this impact pressure in sea-level standard air, so invert the compressible pitot relation at sea level: qc = P₀[(1 + 0.2(CAS/a₀)²)3.5 − 1]. The 0.2 is (γ−1)/2 and the 3.5 is γ/(γ−1) for air with γ = 1.4. Impact pressure depends on calibrated airspeed alone — altitude does not enter.
Second, turn impact pressure into Mach number using the actual static pressure. The same relation runs backwards at altitude: M = √(5[(qc/p + 1)2/7 − 1]). This is where altitude enters. The pressure p comes from the standard atmosphere at your pressure altitude, using the power law below the tropopause and the isothermal exponential above 36,089 ft.
Third, multiply Mach by the local speed of sound. Sound speed depends on temperature alone: a = a₀√(T/T₀), with T in kelvin. That is why you need the outside air temperature and why two aircraft at the same flight level and the same Mach number fly at different true airspeeds on different days.
The familiar cockpit rule — add 2% per 1,000 ft — is a straight line fitted to a curve. On a standard day it runs high through the low and middle altitudes (at 15,000 ft it gives 195 kt against a true 188 kt for 150 kt CAS) and low in the mid-thirties. Use it for a mental check, not for a fuel plan.
Worked example: 150 kt CAS at 8,000 ft, −1 °C
A typical cross-country cruise on a near-standard day.
- Static pressure. 8,000 ÷ 145,441.6 = 0.055002, so p = 1013.25 × (1 − 0.055002)5.255885 = 1013.25 × 0.742805 = 752.65 hPa.
- Impact pressure. CAS/a₀ = 150 ÷ 661.4788 = 0.226768. Squared and multiplied by 0.2 gives 0.010285. Then 1.0102853.5 = 1.036462, minus 1 is 0.036462, times 1013.25 gives qc = 36.947 hPa.
- Mach number. qc/p = 36.947 ÷ 752.65 = 0.049089. Add 1 and raise to 2/7: 1.0490890.285714 = 1.013786. Subtract 1, multiply by 5, take the root: M = √0.068931 = 0.2625.
- Speed of sound. T = −1 + 273.15 = 272.15 K. a = 661.4788 × √(272.15 ÷ 288.15) = 661.4788 × 0.971842 = 642.85 kt.
- True airspeed. TAS = 0.2625 × 642.85 = 168.8 kt.
- Cross-check. ρ = 75,265 ÷ (287.053 × 272.15) = 0.9634 kg/m³, so σ = 0.7865 and the incompressible form gives 150 ÷ √0.7865 = 169.1 kt. The 0.3 kt gap is the compressibility correction, and equivalent airspeed is 168.8 × √0.7865 = 149.7 kt.
The 2% rule would have said 150 × 1.16 = 174 kt, about 3% fast. Over a 300 NM leg that is a nine-minute error in the wrong direction.
Reading the four numbers
True airspeed is your speed relative to the air mass and the only speed that belongs in a navigation calculation. Feed it and the wind into the wind correction angle calculator or the ground speed calculator, then take the groundspeed to flight time and ETA.
Mach number matters as soon as you are fast enough for compressibility to bite. Light aircraft rarely exceed M 0.35 and can ignore it. Turboprops cruise around M 0.45-0.55, business jets and airliners at M 0.72-0.85, and above about M 0.80 local supersonic flow appears over the wing. Every jet has a maximum operating Mach (MMO) alongside its maximum operating speed (VMO); in the climb you hit VMO first and MMO later, and the altitude where they meet is the crossover altitude.
Equivalent airspeed is what the airframe's structure responds to, because it is the true measure of dynamic pressure. In practice pilots fly calibrated airspeed and let the difference be handled in the certification, but if you are computing loads or comparing a flight-test result with a wind tunnel, equivalent airspeed is the currency.
Density ratio tells you how thin the air is: 78.7% in the example. It also predicts your engine's mass flow, and it is directly related to density altitude, which is the same information expressed as an altitude.
A practical note on temperature: at high speed the probe measures total air temperature, which is above static air temperature by roughly a factor of (1 + 0.2M²) times the static value in kelvin. At M 0.82 that is a ram rise of around 30 °C. If your gauge reads total temperature, subtract the ram rise before entering it here.
True airspeed for 150 kt calibrated, standard day
| Pressure altitude | ISA temperature | Static pressure | True airspeed | 2% rule |
|---|---|---|---|---|
| Sea level | 15.0 °C | 1013.3 hPa | 150.0 kt | 150 kt |
| 5,000 ft | 5.1 °C | 843.1 hPa | 161.4 kt | 165 kt |
| 10,000 ft | −4.8 °C | 696.8 hPa | 174.1 kt | 180 kt |
| 15,000 ft | −14.7 °C | 571.8 hPa | 188.2 kt | 195 kt |
| 20,000 ft | −24.6 °C | 465.6 hPa | 204.0 kt | 210 kt |
| 25,000 ft | −34.5 °C | 376.1 hPa | 221.7 kt | 225 kt |
| 30,000 ft | −44.4 °C | 300.9 hPa | 241.7 kt | 240 kt |
| 35,000 ft | −54.3 °C | 238.4 hPa | 264.2 kt | 255 kt |
The rule of thumb runs high up to about 30,000 ft and low above it. On a non-standard day, add roughly 1% of true airspeed for every 5 °C the air is warmer than standard.
Mistakes that put true airspeed out by ten knots
- Feeding indicated airspeed straight in. Apply the position-error correction from the flight manual first. At low speed and high angle of attack the error on a light single is commonly 3-5 kt, and it is largest exactly where the margins are smallest.
- Using indicated altitude instead of pressure altitude. On a low-pressure day the two differ by hundreds of feet. Set 29.92 and read the altimeter, or use the pressure altitude calculator.
- Entering total air temperature at jet speeds. Ram rise inflates the reading, which inflates the speed of sound, which inflates true airspeed. Subtract the ram rise or use the static temperature output of the air data computer.
- Assuming true airspeed always exceeds calibrated. It does not. On a very cold day below about 2,000 ft the air is denser than sea-level standard, and true airspeed comes out below calibrated.
- Trusting the 2% rule at flight levels. It is a linear fit to a curve. It errs high in the teens and twenties and low above about 30,000 ft.
- Confusing equivalent airspeed with calibrated airspeed on a jet. They diverge by more than 15 kt in the mid-thirties, and that difference is the whole reason VMO and MMO are separate limits.
The speed family and when to use which
Six speeds, one chain: indicated → calibrated → equivalent → true → groundspeed, with Mach hanging off the middle. Fly the aeroplane on indicated. Respect the structure on equivalent. Navigate on true. Land on groundspeed.
Mechanical E6B computers solve this with a rotating window that sets pressure altitude against temperature, which effectively computes the density ratio and applies the incompressible relation. That is why a plastic E6B and this calculator agree closely for light aircraft and diverge for jets: the E6B has no compressibility term. Electronic flight bags and air data computers use the compressible relations shown here.
Once you have true airspeed the rest of the flight plan follows. The wind triangle turns true airspeed into groundspeed and a heading — see the wind correction angle calculator. Groundspeed and leg distance give time en route, and time en route with fuel flow gives fuel required. If your route is long enough to need a great-circle track, start with the great circle distance calculator.
One planning caveat: true airspeed rises with altitude at constant calibrated airspeed, but engine power falls, so the altitude that maximises range is a compromise rather than simply the highest you can reach. Consult the cruise performance tables in the flight manual, which give true airspeed and fuel flow together for each power setting and altitude.