Two systems, one line
An azimuth is a single number from 0 to 360, measured clockwise from north. It is what a total station reads, what a coordinate geometry routine wants, and what you can add and subtract without thinking. A quadrant bearing is a letter, an angle no larger than 90 degrees, and another letter: it says start at north or at south, swing this many degrees, toward east or toward west. It is what deeds are written in, what old plats carry, and what a surveyor says out loud.
Neither is going away. Deeds outlive instruments, and a bearing written in 1897 is still the controlling call on a line today. So the conversion gets done constantly, in both directions, usually in a hurry.
The four cases
Going from azimuth to bearing there are four, and only the first is obvious. Between 0 and 90 the bearing is N followed by the azimuth followed by E. Between 90 and 180 it is S, 180 minus the azimuth, E. Between 180 and 270 it is S, the azimuth minus 180, W. Between 270 and 360 it is N, 360 minus the azimuth, W.
The middle two are where the mistakes live. Azimuth 195-30-00 is S 15-30-00 W. Subtract from 360 by reflex and you get N 164-30-00 W, which is not a bearing at all because the angle exceeds 90. Take the difference from 180 without minding which way round it goes and you get 15-30-00 attached to the wrong letters, S 15-30-00 E, which is a perfectly ordinary bearing pointing at a completely different part of the map. Nothing in the arithmetic complains about that one.
Going the other way is easier because the letters tell you what to do. N and E: the azimuth is the angle. S and E: subtract from 180. S and W: add 180. N and W: subtract from 360.
Back bearings
Run a line from A to B and book the bearing. Stand at B and look back at A and you are looking down the same line the other way. The back azimuth is the azimuth plus or minus 180, whichever keeps you inside 0 to 360. The back bearing is easier still: keep the same angle and flip both letters. N 26-10-00 E going out is S 26-10-00 W coming back. That works in all four quadrants with no cases and no arithmetic, which is why it is the version worth carrying in your head.
It is also a field check. Occupy the far end of a line you have already run, sight back, and the direction should be the back bearing of what you booked. If it is not, something moved or something was written down wrong, and finding out at the far end of one line is much cheaper than finding out at the end of the traverse.
Degrees, minutes and seconds
Sixty seconds to a minute and sixty minutes to a degree. The awkwardness is that 26-10-00 is not 26.10 degrees, it is 26.1667, and a calculator that is handed 26.10 will happily produce an answer about seven hundredths of a degree wrong. Over a thousand feet that is 1.2 feet sideways, which is enough to matter on a boundary and not enough to look obviously wrong on the page.
One second of arc subtends about one part in 206,265, so it is five thousandths of a foot at a thousand feet, half a foot at twenty miles, and a whole foot only at about thirty-nine miles. One minute is sixty times that: 0.29 feet at a thousand feet. Those two figures explain the whole shape of the profession. Suburban legs of a few hundred feet are barely troubled by a second, so a twenty-second instrument does ordinary property work; a control network spanning a county is nothing but long lines, and there a second is the whole budget.
Declination, convergence and why this page holds neither
A magnetic compass points at the magnetic pole, which is not the geographic one and does not stay still. The angle between them at your position is the declination, it differs from place to place, and it drifts year on year. A grid north on a plane coordinate system is a third direction again, differing from true north by the convergence, which depends on where you are relative to the central meridian of the projection.
Every one of those is a value with a source: a declination model you run for a specific place and date, the convergence printed on a control sheet, the parameters of a specific projection. This page will apply whatever number you type and will not invent one, because a declination that is right for one county and one year is wrong for the next county and wrong again in five years, and a page that quoted one would be handing you a plausible wrong answer. The field is there, the sign convention is stated, and the value is yours to bring.
It is also why old deed bearings and modern readings disagree. A bearing recorded as magnetic in 1880 was correct then. Read the same line with a compass today and the number is different by however much the declination has moved, and neither reading is a mistake. Reconciling the two is part of what retracement surveying is, and it is not arithmetic.
Questions people ask
How do I convert an azimuth to a bearing?
Four cases. From 0 to 90 the bearing is N, the azimuth, E. From 90 to 180 it is S, 180 minus the azimuth, E. From 180 to 270 it is S, the azimuth minus 180, W. From 270 to 360 it is N, 360 minus the azimuth, W. The two middle cases are where errors happen, because the letters rather than the number tell you which subtraction applies. Azimuth 195-30-00 is S 15-30-00 W.
What is a back bearing?
The direction of the same line looked at from the other end. Keep the angle exactly as it is and flip both letters: N 26-10-00 E becomes S 26-10-00 W. In azimuths it is plus or minus 180, whichever keeps the answer between 0 and 360. It is worth using as a field check — occupy the far end of a line you have run, sight back, and the reading should match.
Is 26-10-00 the same as 26.10 degrees?
No, and this is a common and expensive slip. Ten minutes is ten sixtieths of a degree, so 26-10-00 is 26.1667 degrees. Treating it as 26.10 puts the direction out by about seven hundredths of a degree, which is 1.2 feet sideways over a thousand feet — enough to matter on a boundary and not enough to look wrong on the page.
What declination should I use?
Whatever your own source gives for your position and your date. Declination varies across the country and drifts year on year, so a value is only valid for one place and one time. Run a declination model, or take the figure from the control your work is based on. This page applies the number you type and holds none of its own, because a stated value would be wrong for most readers and would look authoritative anyway.
Why does an old deed bearing not match my compass?
Often because the deed bearing is magnetic and was correct for the declination of its own year, which has since moved. It can also be a different reference entirely — an assumed meridian, a line of an earlier survey, or grid north on a projection. Reconciling a record bearing with a modern one is part of retracement work and involves the record and the monuments, not just a rotation.
How much is one second of arc worth on the ground?
One part in 206,265, which is easier to use than it looks: about 0.005 feet at 1,000 feet, half a foot at twenty miles, and a full foot at roughly thirty-nine miles. At a hundred feet it is half a thousandth of a foot, which no tape resolves. One minute of arc is sixty times as much, about 0.29 feet at 1,000 feet. That scaling is why short property legs tolerate a coarse instrument and long control lines do not.