Quick-reference tools for flight planning and airwork calculations
Flight planning reference →Every result on this page comes from a formula you can work on paper, on a whizz-wheel, or in your head with a rough mental version. Knowing the maths matters for two reasons: you can sanity-check a number that looks wrong, and you can still plan when the tablet has flattened its battery on the apron.
Resolve the wind vector onto the runway. The angle you need is the difference between the wind direction and the runway heading, and the two components fall straight out of a right-angled triangle:
Worked example. Runway 24 (heading 240), reported wind 280 degrees at 20 kt. The angle is 40 degrees. sin(40) = 0.643, so the crosswind is 20 × 0.643 = 12.9 kt, call it 13 kt from the right. cos(40) = 0.766, so the headwind is 20 × 0.766 = 15.3 kt. Notice the headwind is still the larger of the two even at 40 degrees off; the components only cross over at 45 degrees.
The mental version is the clock code: treat the angle in degrees as minutes on a clock face and take that fraction of the wind speed. 40 degrees is 40/60 = 0.67, and 0.67 × 20 = 13.3 kt, which is within half a knot of the trigonometry. The useful anchors are 30 degrees (half the wind), 45 degrees (seven tenths), 60 degrees (nine tenths) and 90 degrees (all of it).
Two habits worth keeping. Use the gust value, not the mean, when you check the crosswind against your own limit: 280/20G32 on runway 24 gives 32 × 0.643 = 20.6 kt of crosswind in the gust. And remember that a POH figure such as the 15 kt quoted for a Cessna 172S is a demonstratedcrosswind, established by a test pilot on a good day, not a certificated limit.
Air gets thinner with height, so the pitot system under-reads. The workhorse rule of thumb is add 2 per cent per 1,000 ft:
Worked example. 110 KIAS at 8,000 ft. 2 per cent × 8 = 16 per cent, so TAS ≈ 110 × 1.16 = 127.6, call it 128 kt. The rule quietly assumes standard temperature, which is where it goes soft. On a warm day at ISA +15 the density altitude at 8,000 ft is about 9,800 ft, and running the same rule on that figure gives 110 × 1.196 = 131.6 kt. Four knots does not sound like much until you spread it over a three-hour leg, where it is about 12 nm of position error in your dead reckoning.
Below roughly 140 kt in a light single, compressibility is negligible and the gap between indicated and calibrated airspeed is a couple of knots at cruise, rising near the stall. The airspeed calibration table in Section 5 of the POH gives the real numbers if you want them.
Two steps. First convert field elevation to pressure altitude using 27 ft per hectopascal, then correct for how far the temperature sits from standard using roughly 120 ft per degree Celsius:
Worked example. A strip at 1,200 ft elevation, QNH 1005, OAT 28 °C. Pressure altitude is 1,200 + (8 × 27) = 1,416 ft. Standard temperature there is 15 − (2 × 1.416) = 12.2 °C, so the day is ISA +15.8. Density altitude is 1,416 + (120 × 15.8) = 1,416 + 1,896 = 3,312 ft, round it to 3,300 ft. The aeroplane will accelerate, climb and cruise as though the strip were at 3,300 ft, which on a normally aspirated light single typically means a ground roll a quarter to a third longer than the sea-level standard figure. Take the number into the POH performance chart; do not stop at the number.
Fuel is a stack, not a single figure: taxi, trip, contingency, alternate, final reserve, plus any extra you choose to carry. Trip fuel comes from time, and time comes from groundspeed:
Worked example. A 130 nm leg, planned TAS 105 kt, headwind component 12 kt, so groundspeed 93 kt. Time is 130 ÷ 93 = 1.398 hours, which is 1 hour 24 minutes. At 8.5 US gallons per hour that is 1.398 × 8.5 = 11.9 gal of trip fuel. Now build the rest of the stack: 1.0 gal for start and taxi; contingency at 5 per cent of trip fuel is 0.6 gal; a 20 nm diversion to the alternate takes 20 ÷ 93 = 0.215 hours (about 13 minutes) and burns 1.8 gal; and the final reserve under the EASA NCO rules for an aeroplane by day is 30 minutes at normal cruise, so 0.5 × 8.5 = 4.25 gal.
Adding up: 1.0 + 11.9 + 0.6 + 1.8 + 4.25 = 19.6, so uplift 20 US gallons. Fly the same trip at night and the final reserve becomes 45 minutes, or 6.4 gal, pushing the total to about 21.7 gal. Convert if you need to: 20 US gal is 75.7 litres and weighs roughly 120 lb (54 kg) of avgas, which is a real number on the mass and balance sheet, not a rounding.
One warning about burn rates. POH figures assume the mixture is leaned at the quoted power setting. Bumbling around at 2,000 ft full rich can add 15 per cent, and a school aircraft with 24 usable gallons that looked like a comfortable three hours becomes a rather tighter two and a half once you have paid the taxi, the contingency, the diversion and the reserve out of the same tank.