# Mathematically calculate Mach number using CAS and PA WITHOUT flight computer

I am just wondering how to calculate the Mach number from CAS and Pressure Altitude WITHOUT the use of a flight calculator.

E.g Given FL300 (30000ft) and 325kt CAS find and Mach number.

I have always used an old-school flight computer but now I am just curious about what's the math formulas behind it.

You can use the following formulas for conversions (source: aerotoolbox.com):

$$M = \sqrt{5 \left[ \left( \frac{q_c}{P} + 1 \right)^{2/7} - 1 \right]}$$ $$\text{CAS} = a_0 \sqrt{5 \left[ \left( \frac{q_c}{P_0} + 1 \right)^{2/7} - 1 \right]}$$

with $$q_c$$ as impact pressure, $$a_0 \approx 340.3 \, \text{m/s}$$ as sea level speed of sound and $$P_0 \approx 101 \, 325 \, \text{Pa}$$ as sea level pressure.

Solving the second equation for $$q_c$$ gives:

$$q_c = P_0 \left( \left[ \frac{1}{5} \left( \frac{\text{CAS}}{a_0} \right)^2 + 1 \right]^{7/2} - 1 \right)$$

Plugging your example $$\text{CAS} = 325 \, \text{kt} \approx 167.2 \, \text{m/s}$$ in gives:

$$q_c \approx 18 \, 181 \, \text{Pa}$$

At FL300 we have $$P \approx 30 \, 090 \, \text{Pa}$$. With the calculated $$q_c$$ we can now calculate the Mach number with the first equation above:

$$M \approx 0.8502$$