A direction can be written two ways. An azimuth is a single angle measured clockwise from north through the full circle. A bearing is an acute angle from the nearer meridian, north or south, together with the quadrant it falls in. They describe the same line, they convert in one step, and the conversion is responsible for more lost marks and more wrong coordinates than any other routine operation in surveying.
The reason is that a bearing carries information in two places - the number and the two letters - and only one of them is arithmetic. Slip a quadrant and every latitude and departure downstream inherits a sign error that closes nothing and looks like a blunder in a distance.
This page covers the four conversion rules and how to remember them, back directions, why every internal computation should run on azimuths, and the specific mistakes worth building a habit against.
The two notations
- Azimuth
- A direction measured clockwise from north, from 0° to 360°. North is 0°, east 90°, south 180°, west 270°. One number, no letters, no ambiguity. Occasionally an older source measures azimuth from south, which is why the reference meridian belongs in the notes.
- Bearing
- An acute angle from north or from south, toward east or west, written N 42°29′30″ E. The angle is never more than 90°, and the two letters place it in one of four quadrants.
Deeds and plats are written in bearings because a bearing is easier to read aloud and harder to transpose: N 42°29′30″ E and N 24°29′30″ E look different in a way that 42°29′30″ and 24°29′30″ do not. Computation runs on azimuths because the sine and cosine of an azimuth carry their own signs, and a bearing's sign has to be supplied by hand from the letters.
The four rules
NE azimuth = bearing angle
SE azimuth = 180° − bearing angle
SW azimuth = 180° + bearing angle
NW azimuth = 360° − bearing angleRather than memorise four lines, memorise one picture. The bearing angle is always the angle from the nearer of north or south, so ask which meridian you are near and whether you are swinging toward east or west. In the north-east quadrant you swing clockwise from north, so the two agree. In the south-east quadrant you swing counter-clockwise from south, so the azimuth is 180° minus the bearing. The remaining two follow by symmetry.
| Azimuth | Bearing | Quadrant | Latitude sign | Departure sign |
|---|---|---|---|---|
| 42°29′30″ | N 42°29′30″ E | NE | + | + |
| 118°41′30″ | S 61°18′30″ E | SE | − | + |
| 205°44′30″ | S 25°44′30″ W | SW | − | − |
| 261°21′00″ | S 81°21′00″ W | SW | − | − |
| 340°19′30″ | N 19°40′30″ W | NW | + | − |
The last two columns are the reason the conversion matters. A latitude is the distance times the cosine of the azimuth and a departure is the distance times the sine, and both signs come free with the azimuth. Work from bearings and you must apply the signs yourself from the letters - which is exactly where the quadrant gets lost.
Back directions
Every line has two directions, one from each end, and a traverse computation moves between them constantly. In azimuths the reverse is a single operation.
back azimuth = forward azimuth ± 180° (add if under 180°, subtract if over)
back bearing = same angle, opposite quadrantThe bearing form is the easier of the two to get right and the easier to get wrong in a specific way: flipping only one letter. N 42°29′30″ E reversed is not N 42°29′30″ W and not S 42°29′30″ E. Both letters change, always, and a reversal that changes one of them has rotated the line rather than reversed it.
Carrying direction through a traverse
Field angles are turned between courses; directions are computed from them. With deflection angles the arithmetic is trivial - add a right deflection to the previous azimuth, subtract a left one - and it comes with a closure check that costs nothing.
| At | From azimuth | Deflection | To azimuth |
|---|---|---|---|
| B | 42°29′30″ | 76°12′00″ R | 118°41′30″ |
| C | 118°41′30″ | 87°03′00″ R | 205°44′30″ |
| D | 205°44′30″ | 55°36′30″ R | 261°21′00″ |
| E | 261°21′00″ | 78°58′30″ R | 340°19′30″ |
| A | 340°19′30″ | 62°10′00″ R | 42°29′30″ |
The deflections sum to 360°00′00″ exactly, which is the geometric condition for a closed traverse turned consistently in one direction: the total change in heading around any closed loop is one full turn. Interior angles satisfy the parallel condition, summing to (n − 2) × 180°, which for five sides is 540°. Whichever angle type your notes carry, check the sum before computing a single latitude.
The mistakes worth guarding against
- Flipping one letter on a back bearing. Both letters change or neither does.
- Applying the SE rule to a NW direction. The tell is a bearing angle that comes out negative or over 90° - arithmetic that cannot be a bearing is telling you the quadrant is wrong.
- Reading a deed bearing and dropping the quadrant into the notes as an azimuth. S 25°44′30″ W is azimuth 205°44′30″, not 25°44′30″, and the resulting course runs the opposite way across the parcel.
- Forgetting that a bearing of exactly N 90°00′00″ E is due east and is more honestly written as azimuth 90°. Directions on the cardinals have two valid bearing forms and neither is wrong, so state one and be consistent.
- Mixing a magnetic bearing from an old deed with grid azimuths from GNSS. They differ by declination plus convergence, both of which vary with place and, for declination, with date. A direction is meaningless without its reference meridian.
None of these is difficult. All of them are common, and all of them are cheap to eliminate with the same discipline: one conversion at the start, one at the end, azimuths in between, and a closure check on the angles before anything else is computed.