Basic math for pilots: does it still matter?

I was on a trip a few weeks ago with a “flow-through” first officer who had been on line with the mainline company for only two months. For the past many years he had been flying the smaller jets for his particular DBA (Doing Business As) company where, obviously, he did the same work as we do on the mainline. And, not surprisingly, he did very well during our entire four-day trip. During the 23 hours we spent crisscrossing the country several times over those four days, we had ample opportunity to get acquainted and talk about one thing and another.

One of the things we discussed was the performance level of many of the new first officers coming on line at the DBAs from the myriad of large universities and flight schools around the country. I’m considered pretty much an old-timer now (less than three years before retirement), and I don’t know very much about all the new training procedures and requirements that the youngsters have to endure in order to eventually get to where I am – the left seat of a large commercial jet. I keep my instructor’s license current, but I’m not heavily involved in teaching anymore.

Cessna on final
Can you fly a safe approach when the weather is good? Sounds easy, but some airline pilots struggle with it.

Anyway, this particular young man was lamenting the performance of several of the new first officers he flew with at the DBA company who were fresh out of these big schools. He noted that each of them could fly instruments very well, and all of them could hand-fly rock-steady ILSs and other instrument approaches down to minimums – cool as the center seeds of cucumbers. From what I’ve heard about the modern training, I expected to hear such.

I was a little shocked, however, when he told me that most of the new-hires came completely unglued when forced to execute visual approaches – especially when cleared for such approaches while still quite high and many miles from the field. He said his flights were often forced to miss the first attempts at visual approaches and go around because of the airplanes being much too high on their profiles; I wondered to myself how such a systemic problem could exist in this computerized age. After some additional discussion, my first officer and I determined that the problem must stem from a lack of understanding of the need of constant situational awareness and simple mathematics.

When training for our Private Pilot licenses however many years ago, we all learned the basics of time, distance, and rate equations, and I won’t waste your time here revisiting them. And, operating a light, single-engine aircraft at relatively low speeds requires only the use of those rudimentary skills in order to be proficient and accurate in lateral and vertical navigation.

On the other hand, when operating a jet aircraft, or any high-speed aircraft, you should always back up those fancy vertical navigation computers with a running mental calculation of your descent profile. What kind of mental calculation am I speaking of? I am speaking of the basic “3-to-1” rule.

In the operation of a large jet aircraft, it is normal to utilize a 3-degree descent profile – the same as most instrument approaches. Why do we use three degrees? Three degrees fits the operation. It’s comfortable – the flight attendants don’t have to work so hard pushing and pulling the drink carts up and down the aisle. Three degrees allows for optimal pressurization performance; it’s what all those brainy engineers (much smarter than me), figured out years ago. What exactly is meant by a “3-degree descent profile?” I mean a 3-degree slope off the horizontal run. Three degrees equals 300 feet per mile, or nautical mile if you so desire.

There are several ways to figure out a three-degree slope descent. Here’s one example:

Math explanation:

If 45° = 100%, then 3° = 5.24%

1 NM = 6076ft

6076 x 0.0524 = 318.4 ft/NM

The math wizzes among the group online here love to have fun figuring out all the different methods and formulas possible to achieve the optimal numbers you need to be precise in your descent profiles, but there is an easy way that seems to work for those of us who never were so mathematically inclined: simply multiply the distance you are from the landing field by three. Then note your groundspeed, add a zero to it, then divide it by half to arrive at your “in-the-ballpark” rate of descent. Example:

Distance from the field is 30 NMs

Groundspeed is 210 knots.

30 x 3 = 90 (Which means the optimal altitude is 9,000 feet AGL for that location).

210, add a zero = 2,100

2,100/2 = 1,050, or roughly 1,000 feet per minute rate of descent.

Garmin GTN VNAV
VNAV is great on the GPS, but so is a quick ballpark calculation in your head.

When on the final approach leg, the same formula can be used if the glideslope for a specific ILS (or any approach) is three degrees:

At five miles distance, the optimal altitude should be somewhere in the vicinity of 1,500 feet AGL. Groundspeed on the approach is, for instance, 140 kts, then the rate of descent should be around 700 feet-per-minute. If you keep these “ballpark” figures in your head as you approach the airports, you’ll never go wrong.

I am no mathematical genius, as you can plainly see, but I do believe that the large universities and flight schools employ many such folks. And I do believe that these learned men and women should incorporate some type of basic math rules-of-thumb for their undergraduates in their pilot programs because, any time things become non-normal, such as on unexpected go-arounds, the danger of making mistakes increases. And mistakes in aviation, as you already know, can be unforgiving at their worst.

Like I said earlier, I am an old-timer, and I will be retiring in a short time from commercial flying, but these formulas have always worked for us on the line, and I’m sure they’ll remain reliable backups to all the new wizzbang computers coming out in the future.

15 Comments

  • Primary training scares students away from numeracy by demanding absurdly over-precise calculations with an ancient circular slide rule, instead of teaching simple rules of thumb like the ones you mention. If we focussed on teaching these, pilots would be more confident about hitting disconnect later in their flying, when the automation isn’t behaving as expected.

  • I earned my private license and instrument rating in the mid- 1980s, and used that “circular slide rule” heavily during training– mastered it, I thought. I used it occasionally for a few years afterward, but from then on, it occupied my flight bag as a ritual icon until I slipped it into a drawer and forgot about it. An electronic device took its place for a while until it also became dead weight. Rules of thumb, the POH, & simple math did the trick for most of my flying. I hope all airline pilots have those basic tools in their kits, but recent airline mishaps suggest otherwise. Recently, I got out my still-nice circle and found it a jumble and challenging to use. How humbling.

  • A while back we were doing what was going to be a right hand ILS approach to 28R in SFO coming from the north in a767. They used to bring us over the bay at 10,000 then a wide right hand tear drop step descended as we went. ” Were we okay for a visual the controller asked?” We were a little high but I said , “No problem”. My F/O who was doing the flying had his ILS all tuned in but I told him to switch off his ILS and make this a REAL visual approach. I was a beautiful clear but dark night … and I said,” no cheating by looking at my ILS”. He was thunderstruck … I never heard so many excuses in my life as to why while he was able to stay lined up with the runway he was all over the sky in altitude, attitude and power control … oh yes the auto throttles were off as well. This guy had more way more time on the a/c than I did … in fact I’d just checked out on the thing so not just recent graduates but experienced pilots who should know better can’t figure it out and by the way I had him detune the references for the outer and inner fix.

  • Dave – thanks for reminding me of a simple formula that a fellow charter pilot used to use (quite a few years ago). He was in some of our better equipment where he always had a VOR/RNAV DME readout to the destination and I’m pretty sure he used the DME x 3 to get an altitude. As the descent progressed, a closer DME x 3 would give a new lower target altitude of where he should be and compared to the indicated altimeter of where he was, he would know whether he needed to make a slight adjustment to the descent rate. That’s all the math that was necessary.

  • I am a retired FedEx MD-11 Captain and I saw the very same thing on the line with new first officers. I cut my teeth in my early career flying DC-8’s with “steam gages” and you had to constantly work those little math problems in your brain when you were setting up for the descent from cruise altitudes. The new “electric” jets like are typical on the line pre-compute a point of distant and eliminate the need for figuring these things out…mostly. It didn’t take long flying the new jets for your brain to go to mush if you depended entirely on the computers to figure those things out. The lesson here…don’t let that happen. Work those little problems in your head and simply get verification from the computers.

    • Walter – how much harder is it to calculate the descent for heavy iron than it is for a little 4-banger from 8,000-10,000 ft up? I’ve never seen the point of descent calculators, etc. over 1,100 hours in my Cherokee, but I understand that it might be a simpler problem.

  • I have taught this rule of thumb to dozens of pilots since I figured it out in the 80s, but especially since Loran and GPS started offering us updated groundspeeds. It’s applicable from student pilots to ATPs. It works in any size aircraft. On any 3-deg. glideslope in the world. It is the DEFINITION of a stabilized approach! Yet most pilots don’t use it.

    I say it as: “Make the vertical speed 5 times the ground speed.” (That’s the other way to do it that will get you the same answer.) Do it over and over again, all the way down the glide path.

    To practice, fly a localizer-only approach and either disable or ignore the glideslope. If you do it right, you will cross each fix at the right altitude and have a smooth descent all the way to the DA. If you’re in a jet going into somewhere without much traffic, do it from FL300 to 50’ AGL, while decreasing speed as needed. It’s best if you do it with the autopilot on in VS mode and just set and reset the descent rate for any changes in ground speed.

  • Marc, yes, I fly with guys who use the 5 X G.S. method, and it works just fine. You have to adjust your descent point slightly for headwinds or tailwinds – normally not more than a few miles or so. But you’re still making those adjustments off the “3 X altitude” method. Some guys automatically add 10% to that distance: ie, 9,000′ to lose. 9 X 3 = 27. 27 + 2.7 = approx. 30 miles out for a 3 degree slope on descent. If your ground speed is 125 kts, multiply 125 X 5 = 625′ per minute rate of descent.

    If you’re flying a relatively fast single-engine airplane like a Bonanza, you might want to keep this formula in your head as a backup:

    Altitude to lose
    _________________ = Minutes to descend
    Rate of descent (ie, 500′ per minute)

    Minutes to descend X N.M per minute (G.S.) = N.M. point at which to begin descent.

    I had to use this one a lot when I was flying charter passengers in unpressurized twins and singles.

  • If 45° is 100%, 3° should be 6.67%, since it is 1/15th of 45.

    If 60° was 100%, then you would get 3° to be 5% exactly.

    • Engineering tables seems like overkill; this is an article about rules of thumb that you can figure fast when you’re in a busy terminal environment, tired after a long flight, and (in an unpressurised plane) maybe just slightly hypoxic.

      If you really want to be more precise, just take the tangent of the angle of descent—the last part of the grade 7/8 math SOHCAHTOA rule—and you have your slope, either in time or distance.

      For the number of feet of descent per nautical mile (6076 ft), your formula from angle n is

      dist = tan(n) * 6076

      So, rounded to the nearest foot,

      tan(3°) * 6076 = 318 ft/nm (~1:20 slope)
      tan(6°) * 6076 = 639 ft/nm (~1:11 slope)
      tan(10°) *6076 = 1,071 ft/nm (~1:6 slope)

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