Bearing and azimuth converter

A direction on a plane survey is one number, and there are two conventions for writing it down. A deed writes N 34°12′45″ E. A total station reads 34°12′45″. They describe the same line. Every conversion on this page is the arithmetic of moving between those two conventions, plus the two operations — reversing a line and turning an angle — that a traverse is built out of.

Azimuth and bearing are the same direction written twice

An azimuth is a single angle measured clockwise from north, running from 0° all the way round to 360°. It is unambiguous, it sorts, it adds, and it is what a modern instrument and every piece of coordinate geometry works in. Note that the reference is north and the rotation is clockwise, which is not the mathematical convention of counter-clockwise from the positive x axis. That difference is the source of most sign errors in hand-written survey code.

A quadrant bearing is a different encoding of the same information: a meridian letter, an acute angle between 0° and 90°, and a departure letter. N 34°12′45″ E means, starting from north, turn 34°12′45″ toward the east. The angle is never larger than a right angle, so the two letters are carrying the rest of the information. A bearing is therefore three symbols where an azimuth is one, and that redundancy is exactly why deeds use it: a transcription error in a bearing is far more likely to produce an impossible direction than a plausible wrong one.

The four quadrant rules

The conversion depends only on which quarter of the circle the azimuth falls in. These are the four published rules, and they are worth memorising because they turn up in every traverse computation done by hand.

The pattern is easier to hold than four lines suggest. In the northern quadrants the bearing is measured from north, so the rule involves 0° or 360°; in the southern quadrants it is measured from south, so the rule involves 180°. The sign always works out so that the answer lands between 0° and 90°. A bearing angle outside that range means the wrong quadrant was picked.

Back sights, back azimuths and back bearings

Every line has two directions, one for each end you stand on. The direction from A to B is the forward azimuth; the direction from B back to A is the back azimuth. On a plane, and only on a plane, they differ by exactly 180°.

The bearing form of the rule is the more pleasant one to work with, because the angle is untouched and only the two letters flip. That is also why a transposed forward and back bearing is such a common blunder in a deed: the digits are identical, and only the letters betray it. A traverse that misses closure by roughly twice the length of one course has almost always had that course entered backwards.

In the field the same word does double duty. A back sight is the observation onto the previously occupied station, and it orients the instrument before any new direction means anything. Enter the back sight azimuth wrong and every direction in that setup is wrong by the same constant — the traverse still closes beautifully on itself while sitting rotated on the ground.

Which notation goes where

• Deeds, plats and legal descriptions are written in quadrant bearings, and have been for as long as there have been metes and bounds descriptions in English. A record document that reads N 26°10′ E should be quoted that way, not converted.
• Instruments and data collectors work in azimuths, or in angles right from a back sight that resolve to azimuths.
• Computation — latitudes and departures, inverses, forwards, area — is always done in azimuths, because sine and cosine want one angle and not a letter pair.
• Astronomic and geodetic work uses azimuths too, with the extra care that the north being referenced may be true, grid or magnetic, and the three differ.

The practical rule that follows: convert to azimuths to compute, and back to bearings to report. Carrying a bearing through a computation means carrying the quadrant logic by hand at every step, which is work with no reward.

Adding and subtracting angles in DMS

Degrees, minutes and seconds are a mixed-radix system — sixty seconds to the minute, sixty minutes to the degree — so the carries do not behave the way decimal carries do. The reliable method is to convert to decimal degrees, work there, and convert back once at the end. Adding DMS columns directly invites a borrow error in the seconds, and that error is a full minute when it happens.

The second panel above reports every sum twice, wrapped into 0° to 360° and unwrapped. The distinction matters more than it looks. A direction has no meaning outside 0° to 360°, so a computed azimuth should always be wrapped. An arithmetic total is a different animal: when you add the interior angles of a closed traverse and compare the total against the geometric condition, wrapping the total destroys the very thing you are checking.

The difference between the observed sum and the required sum is the angular misclosure, normally quoted in seconds of arc. It is spread equally over the observed angles, because each was turned with the same instrument and the same care.

Why the calculator prints ′ and ″

The minute and second marks in an angle are the prime and double prime, U+2032 and U+2033. They are not the apostrophe and the double quotation mark, which are punctuation that happens to look similar in most fonts. Using the correct characters keeps an angle from being mangled by a text editor that helpfully converts straight quotes into curly ones, and it keeps every angle on this site free of escaped quotes in the underlying content files.