Holding Pattern Teardrop Angle

Instrument Procedures

Eddie sez:

There are times when the best entry for a holding pattern is to fly a teardrop, basically cutting a diagonal line across the holding pattern to make the inbound turn more graceful. The standard pilot approach to this is to pick the TLAR answer, "that looks about right." If you get it too wide you will have to fly an intercept heading to the inbound course and risk running out of protected airspace. If you get it too narrow, you will overshoot, have to reintercept from the opposite side, and once again risk running out of protected airspace. There is a better way.

I first learned about this rule of thumb at the Air Force Instrument Instructor's Course (AFIIC) where we were told it is derived from the 60-to-1 concept. But is that true? Well let's find out. But first, we'll define the rule, provide an example of its application, and then we'll provide a mathematical proof of the rule.

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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.



Figure: Holding pattern teardrop angle, from Eddie's notes.

The AFIIC rule is to multiply the airplane's turning radius by 120 and dividing that by the leg distance of the holding pattern. If, for example, you had a turning radius, TR, of a half mile and the leg distance, L, was 10 miles, your best, no-wind entry angle would be (120) (0.5) / (10) = 6 degrees. This works. But can we prove it?


The trigonometry is straight forward (from the drawing):

θ = atan ( 2 TR L )

We've already seen that TR is equal to the ground speed, in nautical miles per minute, squared, divided by 9. Substituting:

θ = atan ( 2 GS2 9 L )

None of this resembles the AFIIC rule of thumb:

θ = ( 120 TR L )

Running through several examples, we see the AFIIC rule of thumb is never off by more than 6% for most typical holding pattern lengths and speeds you would see in a holding pattern. As 120 knots (2 nm/min), for example, the AFIIC rule of thumb says you would need 10.6° for a 5 nm leg, 5.3° for a 10 nm leg, 3.5° for a 15 nm leg, and 2.6° for a 20 nm leg. The actual answers are 10.0°, 5.0°, 3.4°, and 2.5°.

By substituting typical holding pattern speeds for 120 knots (2 nm/min), 150 knots (2.5 nm/hr), and 180 knots (3 nm/hr) we can come up with three more useful formulas.

At 120 knots:

θ120 ( 50 L )

At 150 knots:

θ150 ( 80 L )

At 180 knots:

θ180 ( 120 L )

Looking for a simpler way to present this:

θ Ground speed (knots) -70 L

This is easier to compute than the AFIIC answer, has just as little to do with the actual math, and is as accurate, so we adopt it . . .

Bottom Line

So what about the claims this rule of thumb is based on 60-to-1? My conclusions:

Rule of Thumb 60-to-1? Trigonometry π
Holding Pattern Teardrop Angle    

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.