In the beginning there was TLAR, "That Looks About Right." We pilots learned from experience and tended to fly based on the lessons we had learned over the years. If, for example, pushing the nose over about 1,000 feet prior to level off worked when screaming through the skies with the VVI pegged but waiting till about 300 feet with a slower climb rate was better, well we remembered that.
The problem with TLAR is that is takes experience. If you don't have experience you have to hope the old heads are willing to teach and that you have lots of time to observe. The other problem is that the list of things you had to memorize became very long. When you made a mistake, TLAR became TARA, "That Ain't Right, Adjust."
In the early eighties the United States Air Force had finally had enough of crashing airplanes as a cost of doing business and started getting serious about fixing the mechanical things that were broken and learning how to pass knowledge on from one generation of pilots to the next. The Air Force Instrument Instructor's Course (AFIIC), known over the years under various names, was charged with imparting all that was known about instrument flight.
At AFIIC we were taught that everything in instrument flight can be traced to the relationship of 60 nautical miles to 1 degree and what happens when you divide just about anything by the number 60.
An engineer will tell you that AFIIC stretches the 60-to-1 concept a bridge too far; the magical 60-to-1 idea doesn't actually explain anything. The rules of thumb do not result from simply multiplying or dividing with multiples of the number 60. The real explanation behind each technique has more to do with trigonometry in most cases.
But even with trigonometry some of the AFIIC rules of thumb appear to be just simply made up. In fact, that was my view until an engineer with a much stronger background in this stuff revealed the secret I was missing: radians. (Thank you to Al Klayton, retired Air Force electrical engineer and civilian pilot!) You see our habit of measuring angles in degrees is based on our need to see 360 of them around the world. All that is fine, but there are many advantages to doing the math in radians.
Some of the techniques are simply the result of years of looking for easy to apply rules and being clever. Trig or folklore? To a pilot none of this matters. What matters are accurate techniques that make flying airplanes precisely easier. And these so-called 60-to-1 rules do just that.
In each case we started with the rule of thumb, provided an example, and then ended with a proof of the concept. You might want to skip the math or you might want to get right into the nitty gritty. In either case, these will be followed by:
The rule of thumb will be enclosed in a red box, suitable for framing. If you don't need any of the theory, you can cut to the chase with a concise list of all of these: Rules of Thumb.
There are no references for this. Most of the rules of thumb have been handed down through the generations, some may have been invented at AFIIC, and I will take credit for a few. The math is just a matter of griding through the numbers; the more difficult math is thanks to Al Klayton. (I am not a mathematician. If you see any errors, please let me know, "Contact Eddie" at the bottom of the page.)
Figure: Eratosthenes method for determining the size of the earth, from NOAA (Public Domain)
Contrary to common mythology, the idea that the earth is round predates Columbus. An early Greek scholar, Eratosthenes (276 BC - 195 BC) knew that the sun shone to the bottom of a well in the town of Syene (present day Aswan) on the summer solstice, and was therefore directly overhead. And yet it was not directly overhead in Alexandria, just 925 kilometers directly to the north.
Eratostenes realized the sun's rays reach the earth in virtually parallel lines because of its distance. He measured the angle from vertical of the sun's rays in Alexandria when they were vertical in Syene to be 1/50th of a circle. He rationalized that the circumference of the earth would be 50 times the distance between the cities. Remarkably, he was accurate to within 0.4%.
Of course we know the earth is not a perfect sphere, it is wider at the equator than it is north-to-south. We will use 21,654 nm (24,902 statute miles) for the purposes of the computations to come.
Nobody really knows why there are 360° in a circle, other that a few hypothesis that all sound about right. Ancient astronomers, perhaps, realized each year seems to repeat itself after about 360 days and that the earth, therefore, moved 1/360th of its path around the sun every day.
Greek astronomer Claudius Ptolemy wrote about grids that spanned the earth in a treatise he called "Geography." He cataloged places he knew of in relation to the equator (north and south) and the Fortunate Islands (east and west). The system of using degrees north and south for latitude remain with us to this day. The system east and west still exist, of course, though based on a different location. (The Fortunate Islands are now the Canary Islands and Madeira.)
More about this: Navigation / Latitude and Longitude.
The fundamental 60-to-1 theory comes from the following:
Of course this is off by 0.25%. Close enough!
Figure: 60-to-1 becomes 60 nm at 1° becomes 1 nm, from Eddie's notes.
Any international pilot worth the title knows that 1 degree of latitude equals 60 nautical miles, as proven above. From there we come up with the 60-to-1 theory itself.
The theory tells us that 60 nm horizontally becomes 1 nm vertically, at 1°
We also know that 1 nautical mile equals 6,076 feet.
Which leads us to:
Figure: 60-to-1 becomes 60 nm at 1° becomes 1 nm, from Eddie's notes.
If we divide both sides by 60, we aren't changing the equality so the equation remains true:
We certainly can't read 1 foot on an altimeter, and certainly not 1.27 feet. So the 60-to-1 vertical flight rule becomes:
When you are dealing with the distance travelled along an arc, this is certainly true. But other than flying an arc around a point for an instrument approach, airplanes rarely deal with arcs. Most aviation math has more to do with the right triangle . . .
Photo: "Pay no attention to the man behind the curtain," from The Wizard of Oz.
The classic approach to teaching 60-to-1 is to illustrate the division problem shown above and say that is how it works. Don't look at the man behind the curtain! If you want to understand many of the 60-to-1 rules of thumb, however, simple division isn't going to cut it. A little trigonometry is in order. I promise to make this as painless as possible . . .
Figure: Circle trigonometry, from Stephen Johnson (Wikipedia)
The Greek word for triangle is "trigonon" and from that we get the study of triangles, trigonometry. It is a subject that makes many pilots wince. In fact, you could argue many pilots became pilots because their high school math classes convinced them they should do something fun for a living, rather than spend their days writing formulas and drawing three angles surrounded by three connected lines. That is unfortunate; much of aviation is based on trigonometry.
When you constrain one of the angles in a triangle to being precisely 90°, a right angle, you can learn a lot about the other parts of the triangle with relative ease. If you draw a circle around the triangle with one point at the center and another at the circumference, the tangent of the circle intersecting the triangle has a few interesting properties.
We draw triangles and label the sides with lower case letters. The angle that is opposite that side is labeled by the same letter, in upper case. Just to make things a bit more confusing, we often label the angles using letters of the Greek alphabet, the most common being the letter theta, θ.
The tangent of a triangle is found by dividing the side opposite that angle by the side adjacent to that angle. In a classic mathematical sense, the answer would be presented in radians (of which there are 2π in a circle) but for most uses degrees are preferred.
For example, let's say we have a triangle ABC where a = 1 and b = 2. Using a scientific calculator we see that the tangent of A = 0.1. Now that hardly seems useful, does it?
We can make this function more useful if we could solve for A. This is known as an "inverse function" and the solution for A, in this case, is called the "arc tangent." It can be written as arctangent(A), arctan(A), atan(A), or more properly, tan-1(A).
Our example becomes A = arctan( a / b ) = arctan ( 1 / 2 ) = 27°.
So that is all you really need to know. Just keep in mind this formula converts two sides of a right triangle into an angle. The rest is easy. More about this: Basic Aerodynamics / Trigonometry for Pilots.
This section is courtesy Al Klayton.
Figure: A pie wedge of a circle, from Eddie's notes.
Just like a distance D can be measured in feet D (ft) or nautical miles D (nm), an angle θ can be measured in degrees θ (deg) or radians θ (rad). Like degrees, a radian is defined in relation to the properties of a circle. In particular, an angle θ (rad) is defined as the ratio of the length of a section of the circles’ circumference (arc length S) to the radius R as shown in the figure.
Since we are going to be talking about angles with the Greek symbol theta, θ, with two different units, we will apply a subscript to differentiate between angles measure in radians, θRadians, and degrees, θDegrees.
If the length of S happens to equal R, then
θRadians = S/R = R/R = 1 radian.
If S is twice the length of R, then
θRadians = S/R = 2R/R = 2 radians.
Now if we let S increase to the length of the circle’s circumference then
S = 2πR
and θRadians = S/R = 2πR/R = 2π for a full circle.
So we conclude a complete circle represents 2π radians. But we also know a circle represents 360 degrees. Thus
360 degs = 2π rads
Or 1 deg = 2π/360 rads = π/180 rads, and conversely
1 rad = 360/(2π) = 57.3 degs.
We now can convert back and forth between degrees and radians (like between feet and nm):
Figure: Radians (small angle approximation), from Eddie's notes.
For small angles it is often useful to approximate that the sine or tangent of the angle θRadians is equal to the angle itself, in radians. In other words, sin(θRadians) = tan(θRadians) = θRadians.
Consider the diagram of the circle and right triangle, which is one way to visualize the small angle approximations Sin(θ) = Tan(θ) = θ in radians. CB and AB represent line segments.
Tan(θ) = AB/R but for small θ we can say AB ≈ S, so we have:
Tan(θ) = S/R and by definition S/R is θ in radians.
Likewise we have:
Sin(θ) = AB/CB but for small angles AB ≈ S and CB ≈ R, so we have:
Sin(θ) = S/R and again S/R is θ in radians. Another conclusion is that for small angles:
Tan(θ) = Sin(θ) although the percent error is a bit different.
Al concludes, "There are more mathematically rigorous ways to justify these approximations, but I thought this might provide a little “easy insight”.
So there you have it, eleven rules of thumb that will help you fly instrument procedures with greater accuracy and less guesswork. You don't need any of the math but I've presented it in the associated links to show there is science behind the art. But keeping a list of the rules of thumb may pay dividends in your operational flying.
Turn radius for a 25° bank angle = (nm/min)2 / 9.
Circling Approach, 90° Offset.
To provide circling offset when approaching a runway at 90°, overfly the runway and time for 20 seconds (Category D) or 15 seconds (Category C) before turning downwind.
Circling Approach, from Opposite Runway.
To provide circling offset when approaching from the opposite runway, turn 30° away from heading, time for 66 seconds (Category D) or 53 seconds (Category C), and then turn to parallel the runway on downwind.
Bank Angle for Arc Approach.
The bank angle required to fly an arc is equal to 30 times the aircraft's turn radius (nm) divided by the arc's radius (nm from the station). At low arc distances, this formula tends to be too high.
The distance traveled along an arc is equal to the arc radius times the arc angle divided by 60.
Holding Pattern Teardrop Angle.
A holding pattern teardrop angle can be found by subtracting 70 from the airplane's ground speed (in knots) and dividing the result by the holding pattern leg's distance.
It takes 100 feet vertically to climb or descend at a 1° gradient in 1 nautical mile.
It takes 200 feet at 2°, 300 feet at 3°, and so on.
Flight levels divided by nautical miles equals gradient.
Nautical miles per minute times descent angle times 100 gives vertical velocity in feet per minute.
Top of Descent (3°).
Start descent at three times your altitude to lose in thousands of feet to achieve a three degree gradient.
Top of Descent (2.5°).
Start descent at four times your altitude to lose in thousands of feet to achieve a 2.5 degree gradient.
Visual Descent Point.
A Visual Descent Point is found by subtracting the touchdown zone from the Minimum Descent Altitude and dividing the result by 300.
Portions of this page can be found in the book Flight Lessons 1: Basic Flight, Chapter 24.
Portions of this page can be found in the book Flight Lessons 2: Advanced Flight, Chapters 3, 11, 13, and 17.
So what about the claims these rules of thumb are based on 60-to-1? My conclusion: No, none of them can be correctly called a result of the 60 to 1 relationship. Does that matter? No, not really. If it helps you remember the rules of thumb, good enough.
|Rule of Thumb||60-to-1?||Trigonometry||π|
|Circling Approach 90° Offset||✓|
|Circling Approach From Opposite Runway||✓|
|Bank Angle for an Arc Approach||✓|
|Holding Pattern Teardrop Angle||✓|
|Top of Descent (3°)||✓|
|Top of Descent (2.5°)||✓|
|Visual Descent Point||✓|
60-to-1 — Rule of thumb is based on the mathematical relationship of a 360° circle and/or 6076' to 1 nm.
Trigonometry — Rule of thumb is based on the relationship to a right angle and the derived trigonometric functions.
π — Rule of thumb is based on the relationship of a 360° circle, the number π, and/or 6076' to 1 nm.