The distance traveled along an arc is equal to the arc radius times the arc angle divided by 60.

Eddie Sez:

Here is a 60-to-1 rule that might be better called 360-to-1. Back when I was responsible for arriving at a red carpet plus or minus 5 seconds, knowing the distance travelled along an arc was critical. These days? Not so much. But say you are flying an instrument approach with an arc that allows you to descend gracefully along the arc or dive and drive, knowing the distance can help you judge your descent.

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.


Definition

The distance traveled along an arc is equal to the arc radius times the arc angle divided by 60.


Example

Figure: PHTO VOR/DME Runway 26 Planview, from FAA AL-756, 4 Feb 2016.

In the arc approach into Runway 26 at Hilo (PHTO), for example, you can be cleared for the approach from 5,000 feet at the IAF. You need to descend to 1,800 feet by the time you roll out on final. Is that a speed brakes, stomach churning dive? Or is it a gentle, tropical descent, hardly noticeable from the paying seats? With the aid of this rule of thumb you know that you will have to travel from the 079° radial though 360° (79°) and then from the 360° radial to 310° for another 50°. So you are crossing 79 + 50 = 129°. The rule of thumb tells us we have (11) (129) / 60 = 24 miles to lose 3,200 feet. No big deal.


Proof

Figure: Arc distance, from Eddie's notes.

The AFIIC rule of thumb is this: the distance traveled along an arc is equal to the arc radius times the arc angle divided by 60.

But is that true?

We know, by definition:

Circumference = 2 π r

Circle's angular circumference = 360°

Proportion of Circle = θ 360

Therefore:

d = ( 2 π r ) ( θ 360 ) = r θ 57.30

The exact answer is the arc radius times the angle divided by 57.30. The AFIIC answer is certainly easier to remember and compute and is only in error by 4.5%. We should, therefore, adopt it:

Engineer Al Klayton offers an even better way to look at this, provided you do the math in radians (as explained here: Rules of Thumb / Radians).

θRadians = S/R, so:

S = R (θRadians) which in DME arc language is:

S(DME arc length) = R(DME arc radius) x θ(arc angle in degs) x π/180(conversion from degs to radians)

So if we approximate π as 3 we get:

DME arc length = DME arc radius x DME arc angle/60


More About This:

For more about this rule of thumb, see: Flight Lessons / 60-to-1.


Book Notes

Portions of this page can be found in the book Flight Lessons 2: Advanced Flight, Chapter 11.


Bottom Line

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

Rule of Thumb 60-to-1? Trigonometry π
Arc Distance    

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.