Pilot Math: How much math do you need to know?
A recent post on Aviation Stack Exchange asked whether pilots need to be good at maths. The short answer is that you don’t really need any advanced math skills—although they sometime help to understand things—but you do need to have good arithmetic skills, an understanding of compass points, and an understanding of how graphs and charts work. Here are a few quick examples to show what I mean.
Headwind or Tailwind?
(In the US, runways and wind direction use magnetic direction. In Europe, they often use true direction.) Suppose you are flying to a field without a control tower and one runway that is labelled 11/29. First you listen to the weather broadcast (ATIS) to find out the wind direction and intensity. Suppose that it is 330 at 25. Do you land on runway 29 or runway 11?
Take the wind direction and subtract the runway heading. For Rwy 29 you get 330 – 290 = 40. The wind is coming at you from 40° to the left of the runway. For Rwy 11 you get 330 – 110 = 220. That number is bigger than 180, so it is a tailwind. You don’t usually want to land with a tailwind, so Rwy 29 is the one you will pick.
In this case, the wind is coming at you from 220° to the left of the runway. But that’s probably not the best way to think about it. Subtract 180° from 220° and you get 40° from behind—a tailwind.
Cross-wind Component
Then you have to determine whether the wind is too strong to land on Rwy 29. Airplanes have what is called a maximum demonstrated crosswind—the maximum amount of crosswind that a skilled pilot can land in. You don’t have to know any trigonometry to know how the crosswind component is calculated. There is a chart for figuring out the crosswind component, or you can use a rule of thumb that gets you close enough. If you think of the wind direction in terms of a clock face, you can determine the crosswind component. When the wind is 15° from the runway then imagine a clock face at 15 minutes past the hour. That’s one-quarter of an hour and the crosswind component is one-quarter. A wind at 30° is one-half, 45° is three-quarters, etc. Anything more than 60° is essentially all crosswind. So our wind of 40° is ⅔ of the clock face. The crosswind component is ⅔ * 25 or about 17. The maximum demonstrated crosswind for most light airplanes is around 17 kts, so this is going to be a challenging landing.
You don’t need to to know that determining the direction of the other runway uses modulo 36 arithmetic but you need to be able to understand it. You can always look at your heading indicator or VOR indicator to determine the runway labels and To/From directions for VORs.
Glide Range
Another example is figuring out your gliding distance if you have an engine out. Most general aviation aircraft have a glide ratio in the 8:1 to 12:1 range. You need to know how to read the chart in the owner’s manual to determine the glide ratio for your airplane. This post explains more about glide ratios and range.
This means that for every 1,000′ of altitude, they can glide 8,000′ to 12,000′. Suppose you are on a cross-country flight at 5,500′ in my Cherokee that has a 9:1 glide ratio. How far can you glide. We know that 5,500′ is about 1 statute mile, so you can glide 9 statute miles. If you press the nearest button on your GPS, you can see if there any airports in range. However, the distances on your GPS are usually in nautical miles, so you need to know that a nautical mile is about 6,000′ so your gliding range in nautical miles is about 15% less than in statute miles.
Starting Descent
For passenger comfort, you normally descend an unpressurized plane at around 500 feet per minute. As in the example above, when do you need to start your descent to arrive at a 1,200′ pattern altitude 3 miles from the airport?
Doing a little quick subtraction, you need to descend around 3,000′. That’s six minutes (3,000′ ÷ 500’/minute). So how far will you travel in six minutes? You’ll probably travel faster than normal cruise speed since you’ll be in a descent. Let’s call it 120 kts in my Cherokee. That’s 2 miles per minute. So if you start your descent 12 miles + 3 miles = 15 miles from the airport, then you’ll comfortably arrive at pattern altitude when you want to.
Fuel Remaining
Way too many pilots run out of fuel—seemingly because they can’t do simple arithmetic. My Cherokee burns around 8 gallons per hour in cruise. We fill the Cherokee to the tabs so I start out with 33 gallons of fuel. I always want to land with 1 hour reserve. How many hours can I fly? You could pull out your calculator and do (33 – 8) ÷ 8 = 3.125 or figure out that 33 is a little more than 32 and subtracting 8 gives 24 which is an easy multiple of 8.
The 210 has 89 gallons of useable fuel and burns around 18 gallons per hour in cruise. I’m never sure how close to full it is but there is at least 80 gallons of fuel. Now 5 * 18 is 80, so I can conservatively fly four hours before landing.
Summary
As these examples show, knowing a bit about rounding, chart reading, and simple arithmetic is something every pilot needs to know.