The rule and the navigation triangle
The 1 in 60 rule turns a small sideways displacement into an angle without a protractor. It is an approximation built for quick airborne mental arithmetic.
The basic relationship
If an aircraft is 1 nautical mile off the intended line after travelling 60 nautical miles along it, the angular error is approximately 1 degree. Two nautical miles off after 60 nautical miles gives about 2 degrees, and so on.
The three sides you must identify
| Quantity | Meaning | Where it lies |
|---|---|---|
| Distance gone | Distance measured from the last reliable on-track point to the abeam position of the fix | Along the planned track |
| Distance off track | Shortest sideways separation between the fix and the planned track | Perpendicular to the planned track |
| Track error angle | Angle between planned track and track made good | At the last reliable on-track point |
The last reliable point may be the departure point or a later confirmed on-track fix. The line from that point to the present fix is the track made good. The angular difference between that line and the planned track is the track error angle.
A first worked example
An aircraft planned track 100 degrees true. After 60 NM it is fixed 4 NM right of track. The track error is 60 multiplied by 4, divided by 60, which is 4 degrees right. Track made good is therefore approximately 104 degrees true.
Why 57.3 becomes 60
The rule begins with a radian. Replacing 57.3 by 60 makes the arithmetic easy and introduces only a small deliberate error.
The circle derivation
Take a circle of radius 1 unit. Its circumference is 2 multiplied by pi multiplied by radius. Using pi as 3.142, the circumference is 6.284 units. An arc whose length equals the radius subtends 1 radian at the centre.
For a radius of 1 unit, an arc of 1 unit is therefore associated with 57.3 degrees. Scale the whole circle up so its radius is 57.3 units and an arc of 57.3 units still subtends 57.3 degrees. Angle and arc now have a one-to-one numerical relationship.
The operational approximation
Using 57.3 in mental arithmetic is awkward. Rounding it to 60 changes the exact rule into the 1 in 60 rule. The difference is 2.7 parts in 57.3, about 4.7 percent, normally stated as about 5 percent.
| Geometry | Reference | Meaning |
|---|---|---|
| Exact circular relation | 1 in 57.3 | An arc equal to radius subtends 57.3 degrees |
| Pilot approximation | 1 in 60 | Easy proportion for small angles |
| Deliberate rounding error | About 5 percent | Accepted for quick navigation estimates |
The strict relationship is 1 in 57.3, rounded to 1 in 60 for practical mental calculation.Oxford ATPL Book 10, chapter 10
Track error is obtained by proportioning the distance off track to the distance along track.R.K. Bali, Air Navigation, chapter 13
The right-triangle explanation
A pilot uses a right triangle rather than a circular arc. For a small angle, the tangent relationship is close enough to the 1 in 60 proportion.
Opposite over adjacent
In the navigation triangle, distance off track is the opposite side and distance gone is the adjacent side. Exact plane geometry gives tangent of the error angle equal to opposite divided by adjacent.
At an angle of 10 degrees, tangent 10 degrees is about 0.176. With an adjacent side of 60 NM, the exact opposite is 60 multiplied by 0.176, about 10.56 NM. The 1 in 60 rule estimates 10 NM. The answer is close, but not exact.
How the approximation changes with angle
| Angle | Tangent | Exact opposite at 60 units | 1 in 60 estimate |
|---|---|---|---|
| 1 degree | 0.017 | 1.02 | 1 |
| 2 degrees | 0.035 | 2.10 | 2 |
| 5 degrees | 0.087 | 5.22 | 5 |
| 10 degrees | 0.176 | 10.56 | 10 |
| 15 degrees | 0.268 | 16.08 | 15 |
| 20 degrees | 0.364 | 21.84 | 20 |
Two approximations are present. First, 57.3 has been rounded to 60. Second, the circular arc has been represented by the opposite side of a right triangle. The tangent is increasingly non-linear as the angle grows.
Expanding and contracting the triangle
A fix rarely arrives exactly 60 NM after the last on-track point. Similar triangles let the same ratio work at any practical distance.
Similar triangles preserve the angle
If the angle stays the same, doubling both the along-track and off-track sides keeps the triangle similar. An 8 degree error gives about 8 NM off after 60 NM, 4 NM off after 30 NM, or 16 NM off after 120 NM.
| Distance gone | Distance off | Equivalent ratio at 60 | Error |
|---|---|---|---|
| 30 NM | 4 NM | 8 in 60 | 8 degrees |
| 60 NM | 8 NM | 8 in 60 | 8 degrees |
| 120 NM | 10 NM | 5 in 60 | 5 degrees |
Use the formula when the ratio is not obvious
An aircraft is fixed 6 NM right of track after 40 NM. Multiply 6 by 60 to obtain 360, then divide by 40. The track error angle is 9 degrees right.
At 120 NM gone and 8 NM right, the calculation is 8 multiplied by 60, divided by 120. The track error is 4 degrees right. The greater distance gone produces the smaller angle for the same sideways displacement.
Rearranging the rule
The same proportion can find an angle, a lateral displacement, or the distance over which an observed displacement developed.
Find distance off track
A track error of 5 degrees persists for 84 NM. Distance off is 5 multiplied by 84, divided by 60, which is 7 NM.
Find distance gone
An aircraft is 6 NM off track and the track error is 4 degrees. Distance gone is 60 multiplied by 6, divided by 4, which is 90 NM.
Keep the units consistent
Distance off and distance gone must use the same units. Nautical miles are usual, but kilometres or metres also work if both distances use that same unit. The units cancel in their ratio, leaving an angle in degrees.
| Known values | Required value | Rearrangement |
|---|---|---|
| Off and gone | Error angle | 60 x off divided by gone |
| Angle and gone | Distance off | Angle x gone divided by 60 |
| Angle and off | Distance gone | 60 x off divided by angle |
Track made good
Once the angular error is known, it can be applied to the planned track to estimate the track actually flown.
Apply left and right correctly
| Fix relative to planned track | Track error | Track made good |
|---|---|---|
| Right of track | Right | Planned track plus error |
| Left of track | Left | Planned track minus error |
A planned track is 045 degrees true. After 80 NM the fix is 4 NM left of track. The error is 60 multiplied by 4, divided by 80, which is 3 degrees left. Track made good is 045 minus 3, or 042 degrees true.
A planned track is 220 degrees true. After 45 NM the fix is 3 NM right of track. The error is 60 multiplied by 3, divided by 45, which is 4 degrees right. Track made good is 220 plus 4, or 224 degrees true.
Crossing north
Apply the correction as an angle, then normalise the answer into the 000 to 359 degree system. For example, planned track 358 degrees with a 5 degree right error gives track made good 003 degrees.
Using the rule reliably
The arithmetic is short. Most errors come from choosing the wrong point, wrong distance, wrong side, or a geometry too large for the approximation.
A disciplined sequence
- Confirm the last reliable point from which the aircraft was known to be on track.
- Measure distance gone along the planned track from that point to the abeam position of the present fix.
- Measure the shortest perpendicular distance from the fix to the planned track.
- Calculate 60 multiplied by off track, divided by distance gone.
- Label the answer left or right before applying it to planned track.
- Check whether the angle is small enough for the approximation to be sensible.
Common traps
| Trap | Why it is wrong | Correct action |
|---|---|---|
| Using slant distance from start to fix | The formula needs the adjacent along-track distance | Use the planned-track distance to the abeam point |
| Using distance to destination | That creates a closing-angle triangle | Use distance gone for track error |
| Ignoring left or right | The magnitude alone cannot give track made good | Write L or R with every answer |
| Mixing NM and km | The ratio is no longer dimensionless | Convert both distances to one unit |
| Using a large angle blindly | The tangent becomes increasingly non-linear | Use exact trigonometry when greater accuracy is needed |
Boundary of this chapter
This chapter establishes the error triangle and its basic rearrangements. Returning to the original track, flying directly to destination, the double-track-error method, glidepaths and radio-aid applications are separate uses developed in the next two chapters.