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).

Flying a precise bank angle around an arc approach is one of those skills that seemed unnecessary at the time and even less so today. But properly executed it was impressive. We needed to do these to pass checkrides and after a while you developed a feeling ("TLAR" - that looks about right) for what bank angle was needed. But most pilots would bracket the arc. Here are my notes from 1979: Instrument Procedures / Arcs.

If you regularly go someplace where an arc approach is used, such as Nice, France, you should give this a try. The other pilot will be impressed.

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.

Thanks as always to Al Klayton, retired Air Force electrical engineer and civilian pilot for help with the trickier math.

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).

Figure: VOR B Rwy 22L/R, Planview, from Jeppesen Airway Manual, page LFMN 13-2, 19 Jun 2015.

Many modern aircraft have Flight Management Systems that know how to fly an arc around a VOR and you might go an entire career without this skill. Or maybe not. In 2002 I was in the right seat of a Challenger 604 and noted the approach was not in the database and the pilot would have to fly it manually. He looked at the arc and said they would never make him do that. Of course they did. "I don't know how to do that," he said. I flew it from the right seat, explaining as I did. He said it was like flying in the days of the early air mail delivery service. It isn't that hard, see Instrument Procedures / Arcs for my notes from back in 1979.

You can fly the arc by picking a heading that places the bearing pointer on your inside wing tip and then adjusting the heading to bracket the bearing pointer plus 10°, minus 5° of the wing tip. Or you can fly the exact bank angle. In this case, flying at 150 knots (2.5 nm/min) we know we have a turn radius of (2.5)^{2}/9 = 0.69 nm. We then can find the precise bank angle: (0.69)(30)/(20) = 1 degree. Not much, of course.

We already figured a formula to compute radius from bank angle in step 15 of Rules of Thumb / Turn Radius. For the sake of clarity, we'll call the radius resulting from the velocity and bank angle combination the ARCRADIUS:

$\mathrm{ARCRADIUS}=\frac{{V}^{2}}{18.96tan\left(\theta \right)}$We can solve for the bank angle, θ, with a little algebra:

$\theta =atan\left(\frac{{V}^{2}}{18.96\left(\mathrm{ARCRADIUS}\right)}\right)$We also know that our turn radius, TR, is equal to our velocity squared divided by 9. Substituting:

$\theta =atan\left(\frac{\mathrm{TR}}{2.11\left(\mathrm{ARCRADIUS}\right)}\right)$For the purposes of our math here, 2 is close enough to 2.11, so . . .

$\theta \approx atan\left(\frac{\mathrm{TR}}{2\left(\mathrm{ARCRADIUS}\right)}\right)$So, by definition:

$tan\theta \approx \left(\frac{\mathrm{TR}}{2\left(\mathrm{ARCRADIUS}\right)}\right)$For small angles we can say the tan(θ) = θ in radians, but we are dealing with degrees. Therefore:

$tan\left(\frac{\theta \pi}{180}\right)\approx \frac{\theta \pi}{180}$Substituting:

$\frac{\theta \pi}{180}\approx \frac{\mathrm{TR}}{2\left(\mathrm{ARCRADIUS}\right)}$Let's call π ≈ 3. Then it follows . . .

$\frac{\theta 3}{180}\approx \frac{\mathrm{TR}}{2\left(\mathrm{ARCRADIUS}\right)}$A little math follows . . .

$\frac{\theta}{60}\approx \frac{\mathrm{TR}}{2\left(\mathrm{ARCRADIUS}\right)}$And finally we come up it the AFIIC rule of thumb. . .

$\theta \approx \frac{30\mathrm{TR}}{\mathrm{ARCRADIUS}}$This works pretty well until the arc radius gets smaller than 5 or so.

So for our LFMN (Nice, France) example with the 20 DME arc radius off of AZR flying at 150 knots with a turn radius of 0.69 nm, the two formulas produce the following:

$\theta =atan\left(\frac{\mathrm{TR}}{2.11r}\right)$ $\theta =atan\left(\frac{\mathrm{0.69}}{2.11\left(\mathrm{20}\right)}\right)=0.94\mathrm{degrees}$The AFIIC rule of thumb is close:

$\theta =\frac{30\left(\mathrm{0.69}\right)}{\mathrm{20}}=1.04\mathrm{degrees}$So in our Nice, France example it was hardly any bank at all. What about arc that is just 5 DME?

$\theta =atan\left(\frac{\mathrm{0.69}}{2.11\left(5\right)}\right)=3.74\mathrm{degrees}$The AFIIC rule of thumb is off a little bit:

$\theta =\frac{30\left(\mathrm{0.69}\right)}{5}=4.14\mathrm{degrees}$Not really a big deal, plus or minus a degree is pretty good.

For more about this rule of thumb, see: Instrument Procedures / Arcs.

Portions of this page can be found in the book Flight Lessons 2: Advanced Flight, Chapter 13.

So what about the claims this rule of thumb is based on 60-to-1? My conclusion: No, it has more to do with trigonometry.

Rule of Thumb | 60-to-1? | Trigonometry | π |

Bank Angle for Arc Approach | ✓ |

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.