Mach Number Calculator

Your Mach number is how fast you are going compared with the speed of sound in the air around you. Mach 0.78 means 78% of the speed of sound. The speed of sound depends only on the air temperature: it is faster in warm air and slower in cold air. Pressure and altitude don't change it on their own.

This calculator uses true airspeed (TAS), your real speed through the air. It does not use indicated (IAS) or calibrated (CAS) airspeed. To turn IAS or CAS into TAS first, use the true airspeed calculator.

The formula

temperature in kelvin = °C + 273.15
speed of sound a (kt) = 38.967854 × √(kelvin)
Mach = TAS ÷ a, and TAS = Mach × a
ISA temperature (°C) = 15 − 1.9812 × (pressure altitude in ft ÷ 1,000), but never below −56.5 (above 36,089 ft)
ISA temperature with a deviation = ISA + deviation

The number 38.967854 comes from √(1.4 × 287.05287) = 20.0468 m/s per √K, turned into knots (× 3,600 ÷ 1,852). At 15 °C (288.15 K) it gives the standard sea-level speed of sound, 661.48 kt or 340.29 m/s.

Worked example

You are cruising at 450 kt TAS and the outside air is −56.5 °C (the standard temperature above 36,089 ft).

• Temperature in kelvin: −56.5 + 273.15 = 216.65 K.
• Square root: √216.65 = 14.71904.
• Speed of sound: 38.967854 × 14.71904 = 573.57 kt.
• Mach: 450 ÷ 573.57 = 0.78456, so Mach 0.785, just under the Mach 0.8 edge of the transonic band.
• The other way: Mach 0.78 at the same temperature is 0.78 × 573.57 = 447.4 kt TAS.

Mach and TAS by temperature

Speed of sound and the TAS for Mach 0.78 at a few temperatures (ISA values at the altitudes shown):

Why the same Mach is slower in cold air

Airliners cruise at a set Mach number. Because the speed of sound falls as the air cools, holding the same Mach in colder air means a lower true airspeed and a lower ground speed. On the climb, pilots fly a set IAS until the "crossover altitude", where that IAS and the cruise Mach give the same TAS. Above it, they fly the Mach number, and TAS keeps falling as the air cools until the tropopause (about 36,089 ft in ISA), where the temperature stops falling.

Flight regimes

These are the common textbook bands. The exact edges vary by source and by aircraft shape.

• Subsonic: below Mach 0.8. The air flows smoothly everywhere.
• Transonic: Mach 0.8 to 1.2. Some of the air over the wing goes faster than sound and small shock waves form. Airliners cruise at the low end, around Mach 0.78 to 0.85.
• Supersonic: Mach 1.2 to 5, like Concorde (about Mach 2) and fighter jets.
• Hypersonic: above Mach 5, like re-entering spacecraft.

Total air temperature

At speed, the air hitting a temperature probe heats up. In kelvin, TAT = OAT × (1 + 0.2 × Mach²). At Mach 0.785 and −56.5 °C, TAT = 216.65 × 1.12311 = 243.32 K, or −29.8 °C: a rise of 26.7 °C. Use the outside (static) temperature here, not the TAT reading.

Uses dry air, γ = 1.4 and R = 287.05287 J/(kg·K), and the ICAO Standard Atmosphere (15 °C at sea level, −6.5 °C per km up to 11 km, then −56.5 °C to 20 km), checked 2026-10-13. For planning and training: use your aircraft's approved flight manual for operational figures.