Two ways to write the same direction
Every line on a survey has one direction, but the profession writes it two different ways and you will meet both on the same project.
An azimuth is a single angle turned clockwise from north, running from 0° through 360°. It is unambiguous, it sorts numerically, and it drops straight into the coordinate geometry that computes departures and latitudes. Every total station, every GNSS baseline and every COGO routine works in azimuths.
A quadrant bearing names the nearer meridian, gives an acute angle from it, and names the side the line falls on: N 45°30′00″ E. The angle never exceeds 90°. Deeds, metes-and-bounds legal descriptions and older plats are written this way, and in many jurisdictions the recorded description still controls, so you cannot simply convert everything to azimuths and forget the original.
The conversion is arithmetic, not measurement, so it introduces no error of its own — but it introduces plenty of blunders, because each quadrant uses a different rule and two of the four subtract rather than add. A sign error here rotates a whole parcel and closes a traverse on the wrong side of the street.
The four quadrant rules and why they take that form
Picture the compass rose. North is 0°, east is 90°, south is 180°, west is 270°, and azimuth increases clockwise. A quadrant bearing measures from whichever of north or south is nearer, so the rule for each quadrant just re-expresses that measurement on the 0–360 scale.
North-east (azimuth 0° to 90°): the bearing is already measured clockwise from north, so Az = B. N 45°30′ E is azimuth 45°30′.
South-east (90° to 180°): the bearing is measured counter-clockwise from south. South is 180°, so Az = 180° − B. S 67°30′ E is azimuth 112°30′.
South-west (180° to 270°): measured clockwise from south, so Az = 180° + B. S 30°15′ W is azimuth 210°15′.
North-west (270° to 360°): measured counter-clockwise from north, so Az = 360° − B. N 75° W is azimuth 285°.
The pattern is worth memorising as a shape rather than as four rules: you add when the two letters read in the same rotational sense (NE and SW), and subtract when they oppose (SE and NW). Going the other way, from azimuth to bearing, you subtract from or into the nearer of 0°, 180° or 360°.
The back azimuth is the direction you would read looking back along the line, which is exactly 180° away: add 180° and, if the result exceeds 360°, subtract 360°. The reverse bearing keeps the same angle and swaps both letters, because reflecting a direction through the origin flips north to south and east to west simultaneously. N 45°30′ E reversed is S 45°30′ W.
Degrees-minutes-seconds is plain sexagesimal: one degree is 60 minutes and one minute is 60 seconds, so decimal degrees = D + M/60 + S/3600. Going back the other way is where the carry bites — 0.99999° is 59′59.996″, which rounds to 60′00.00″ and must then carry into a whole degree. This calculator performs that carry; many spreadsheets do not.
Worked example: converting a deed course by hand
A deed reads “thence S 67°30′00″ E, 214.86 feet”. You need the azimuth for a coordinate computation, and the back azimuth so the adjoining parcel's description can be checked.
- Convert the angle to decimal degrees. 67 + 30 ÷ 60 + 0 ÷ 3600 = 67 + 0.5 = 67.5°.
- Identify the quadrant. The letters are S and E, so this is the south-east quadrant and the rule is Az = 180° − B.
- Apply the rule. Az = 180 − 67.5 = 112.5°, which written back in DMS is 112°30′00″.
- Back azimuth. 112.5 + 180 = 292.5°. That is under 360, so no reduction is needed.
- Reverse bearing. 292.5° falls between 270° and 360°, so it is the north-west quadrant with angle 360 − 292.5 = 67.5°, giving N 67°30′00″ W — the same 67.5° with both letters swapped, as expected.
- Check with mils, if you need them. 112.5 × 6400 ÷ 360 = 2,000 mils exactly.
With the azimuth in hand, the departure and latitude of the course follow directly: departure = 214.86 × sin 112.5° = 214.86 × 0.923880 = 198.50 ft east, and latitude = 214.86 × cos 112.5° = 214.86 × (−0.382683) = −82.22 ft, that is 82.22 ft south. The signs come out of the azimuth automatically, which is precisely why traverse computations are done in azimuths rather than in bearings.
How to check that a converted direction is right
Sanity-check the quadrant before the arithmetic. An azimuth between 0 and 90 must produce an NE bearing, 90 to 180 an SE bearing, 180 to 270 an SW bearing, and 270 to 360 an NW bearing. If your converted bearing letters do not match the numeric range, you have applied the wrong rule — usually by adding where you should have subtracted.
The bearing angle must never exceed 90°. If a hand conversion gives you N 112° E, you have added instead of subtracted in the south-east quadrant. The calculator rejects an out-of-range bearing angle outright rather than quietly wrapping it.
Forward and back must differ by exactly 180°. On a closed traverse, the back azimuth of each course should equal the forward azimuth of the same line computed from the other end. Any discrepancy is a blunder in the angle book, not an accumulation of measurement error.
Watch which north you are on. Azimuths on a state plane grid are referred to grid north, which differs from geodetic north by the convergence angle — up to a degree or more near the edges of a zone. A magnetic bearing from a hand compass differs from true north by the local declination, which changes with time. Converting between bearing and azimuth does not change the reference meridian, so a magnetic bearing converts into a magnetic azimuth and nothing else.
Old deeds carry old meridians. A bearing written in 1890 was probably magnetic, and declination in much of North America has shifted several degrees since. Retracing such a description means rotating the whole set of courses to fit found monuments, not converting each one in isolation.
Azimuth and quadrant bearing side by side
| Azimuth | Quadrant bearing | Rule applied | Back azimuth | Reverse bearing |
|---|---|---|---|---|
| 0°00′00″ | Due north | — | 180°00′00″ | Due south |
| 30°00′00″ | N 30°00′00″ E | Az = B | 210°00′00″ | S 30°00′00″ W |
| 90°00′00″ | Due east | quadrant boundary | 270°00′00″ | Due west |
| 112°30′00″ | S 67°30′00″ E | Az = 180 − B | 292°30′00″ | N 67°30′00″ W |
| 180°00′00″ | Due south | — | 0°00′00″ | Due north |
| 210°15′00″ | S 30°15′00″ W | Az = 180 + B | 30°15′00″ | N 30°15′00″ E |
| 270°00′00″ | Due west | quadrant boundary | 90°00′00″ | Due east |
| 285°00′00″ | N 75°00′00″ W | Az = 360 − B | 105°00′00″ | S 75°00′00″ E |
| 359°00′00″ | N 1°00′00″ W | Az = 360 − B | 179°00′00″ | S 1°00′00″ E |
Directions falling exactly on a cardinal point are written as “due north”, “due east” and so on, because both quadrant letters would be equally valid.
Blunders this conversion invites
- Applying the north-east rule everywhere. Az = B is correct in only one quadrant out of four. It is also the rule people remember, which is why the other three produce most errors.
- Reading the letters out of order. The first letter names the meridian the angle is measured from, the second names the side. “E 45° N” is not a valid bearing and usually signals a transcription error in the source.
- Losing the seconds carry. Converting 45.9999722° back to DMS naively gives 45°59′59.9″ but rounding the seconds to two places yields 60.00″, which must carry into the minutes and possibly into the degrees.
- Confusing minutes of arc with decimal minutes. 45°30′ is 45.5°, not 45.30°. A deed angle typed straight into a decimal-degree field is wrong by up to 0.4°, which is 7 ft over 1,000.
- Mixing grid, geodetic and magnetic north in one traverse. The conversion preserves whatever meridian you started with; it does not reconcile them.
- Forgetting to reduce a back azimuth past 360°. Adding 180° to 285° gives 465°, which must become 105°.
- Assuming a reverse bearing changes only one letter. Both letters swap: the reverse of N 45° E is S 45° W, not S 45° E.
Where directions go next
A converted azimuth is rarely the end of the job. In a traverse computation, each course's azimuth and distance become a departure (E–W component) and a latitude (N–S component); summing them around a closed figure gives the misclosure, which is distributed by the compass rule or by least squares. In a deflection-angle traverse, you carry azimuth forward from course to course by adding the deflection angle, and the back azimuth is what you start each addition from.
On highway work, the azimuth of the back tangent plus the intersection angle gives the azimuth of the forward tangent, which is the first thing a horizontal curve layout needs before it can place the PC and PT. Earthwork and grading follow the same alignment, so the direction column feeds directly into a cut and fill computation and into the trench alignments behind a trench excavation volume.
Directions expressed as slopes rather than angles turn up constantly on the construction side — a roof pitch is the same arctangent relationship written as rise over run, and converting between the two is the vertical analogue of what this page does horizontally.
Two related conventions are worth knowing. Artillery and land navigation use mils, with 6,400 to a circle in the NATO system, because at that scale one mil subtends roughly one metre at a kilometre. Continental European survey work sometimes uses gons (grads), with 400 to a circle, so a right angle is a round 100. Both are just rescalings of the azimuth.
Key terms
- Azimuth
- A horizontal direction measured clockwise from a reference meridian, running 0° to 360°. Unless stated otherwise, surveying azimuths are measured from north.
- Quadrant bearing
- A direction written as an acute angle from north or south, with the quadrant named by two letters: N 30°15′ W. Sometimes called a compass bearing.
- Back azimuth
- The azimuth of the same line looked at from the other end, exactly 180° from the forward azimuth.
- Convergence
- The angle between grid north and geodetic north at a point. It is zero on the central meridian of a state plane zone and grows toward the zone edges.
- Magnetic declination
- The angle between magnetic north and true north at a place and time. It changes measurably over decades, which is why old magnetic bearings must be rotated rather than converted.
- Metes and bounds
- A legal land description that walks the boundary course by course, each with a direction and a distance, returning to the point of beginning.