A whole circle bearing (WCB) is the angle of a survey line measured clockwise from north, from 0° to 360°. A reduced bearing (RB), also called a quadrantal bearing, is the same direction measured from north or south, whichever is nearer, towards east or west, so it is never more than 90°. To convert, find the quadrant first: up to 90° the RB is N (WCB) E, from 90° to 180° it is S (180° − WCB) E, from 180° to 270° it is S (WCB − 180°) W, and from 270° to 360° it is N (360° − WCB) W. So a line with a WCB of 238°15' has a reduced bearing of S 58°15' W.
That one rule appears in almost every surveying exam in Nepal, from IOE and PU semester papers to the NEC licence and Lok Sewa sub-engineer tests. On site it matters too: a prismatic compass reads WCB, an old surveyor's compass and many textbook problems use RB, and latitudes and departures are easiest to work out from RB. This guide gives the rules, back bearings, how to get included angles from bearings, how to spot and correct local attraction, and a full worked traverse for a plot of about one ropani, with every number checked.
WCB and RB in one picture
Think of the compass circle split into four quadrants by the north-south and east-west lines. WCB always starts at north and goes clockwise all the way round. RB starts at north for the two upper quadrants and at south for the two lower ones, and goes towards east or west. The letters in an RB tell you which quadrant the line is in; the number tells you how far it is from the north-south line.
Conversion rules
| WCB range | Quadrant | WCB to RB | RB to WCB |
|---|---|---|---|
| 0° to 90° | I (NE) | N θ E, θ = WCB | WCB = θ |
| 90° to 180° | II (SE) | S θ E, θ = 180° − WCB | WCB = 180° − θ |
| 180° to 270° | III (SW) | S θ W, θ = WCB − 180° | WCB = 180° + θ |
| 270° to 360° | IV (NW) | N θ W, θ = 360° − WCB | WCB = 360° − θ |
Lines exactly on an axis have no quadrant. A WCB of 90° is due east, 180° is due south and 270° is due west. Write them as "due E" and so on, or as N 90° E if your teacher prefers.
Worked example 1: converting a set of bearings
The first four lines below are the sides of the traverse used later in this article. The last three go the other way, from RB to WCB.
| Given | Quadrant | Working | Answer |
|---|---|---|---|
| WCB 57°45' | I | same as WCB | N 57°45' E |
| WCB 140°00' | II | 180° − 140°00' | S 40°00' E |
| WCB 238°15' | III | 238°15' − 180° | S 58°15' W |
| WCB 323°30' | IV | 360° − 323°30' | N 36°30' W |
| RB S 25°30' W | III | 180° + 25°30' | WCB 205°30' |
| RB N 12°15' W | IV | 360° − 12°15' | WCB 347°45' |
| RB S 80°00' E | II | 180° − 80°00' | WCB 100°00' |
When you subtract degrees and minutes, borrow 60 minutes from the degrees. For 360° − 323°30', write 360° as 359°60', then 359°60' − 323°30' = 36°30'.
Fore bearing and back bearing
The fore bearing (FB) of line AB is read at A looking towards B. The back bearing (BB) is read at B looking back to A. The two directions are exactly opposite, so:
- In WCB: BB = FB ± 180°. Add 180° if the FB is less than 180°, subtract if it is more. FB 57°45' gives BB 237°45'; FB 323°30' gives BB 143°30'.
- In RB: keep the same angle and swap both letters. N 57°45' E becomes S 57°45' W; S 40°00' E becomes N 40°00' W.
In the field the FB and BB of a line should differ by exactly 180°. If they do not, and the compass itself is in order, something near one of the stations is pulling the needle. That is local attraction, covered below.
Included angles from bearings
You often need the angle between two lines at a station, for example to check a closed traverse. With stations lettered clockwise round the figure, the interior angle at B is:
Interior angle at B = BB of AB − FB of BC (add 360° if the result is negative).
If the stations go anticlockwise, reverse it: FB of BC − BB of AB. For a closed traverse with n sides the interior angles must add up to (2n − 4) × 90°, which is 360° for 4 sides, 540° for 5 and 720° for 6. This check works even when the bearings have local attraction, because an error at a station shifts both bearings read there by the same amount and cancels in the angle.
Local attraction: how to find and correct it
A magnetic needle is pulled off north by iron and steel close to the instrument: reinforcement bars stacked on a site, a steel gate, an electricity pole, a parked motorcycle, even a mobile phone or a bunch of keys in the surveyor's pocket. In Nepali towns, where most plots have an RCC house, a steel shutter or a pole nearby, local attraction is common. It affects every bearing read at that station by the same amount.
The method used in most textbooks:
- Tabulate the observed FB and BB of every line and find FB − BB for each.
- Find a line whose FB and BB differ by exactly 180°. Both its end stations are taken as free of local attraction.
- Start from that line. At its forward station the BB is correct, so the bearing of the next line read at the same station needs no change.
- Move to the next station. Compare its observed BB with the correct value (FB ± 180°). The difference is the error at that station. Apply the same correction, with the same sign, to the FB read there.
- Carry on round the traverse. The last BB, read at the starting station, should come out right with no correction. That is your check.
Worked example 2: a four-sided compass traverse ABCD
A plot boundary was traversed clockwise with a prismatic compass. Observed bearings:
| Line | Observed FB | Observed BB | Difference |
|---|---|---|---|
| AB | 57°45' | 237°45' | 180°00' |
| BC | 140°00' | 320°45' | 180°45' |
| CD | 239°00' | 57°45' | 178°45' |
| DA | 323°00' | 143°30' | 180°30' |
Step 1. Only AB differs by exactly 180°, so A and B are free of local attraction. FB of AB = 57°45' and FB of BC = 140°00' (read at B) are correct.
Step 2. At C, the correct BB of BC is 140°00' + 180° = 320°00'. Observed is 320°45', so the error at C is +0°45' and the correction is −0°45'. Corrected FB of CD = 239°00' − 0°45' = 238°15'.
Step 3. At D, the correct BB of CD is 238°15' − 180° = 58°15'. Observed is 57°45', so the error at D is −0°30' and the correction is +0°30'. Corrected FB of DA = 323°00' + 0°30' = 323°30'.
Step 4 (check). Correct BB of DA = 323°30' − 180° = 143°30', which matches the value observed at A. A is free, so the working closes.
| Line | Corrected FB (WCB) | Corrected BB | Reduced bearing |
|---|---|---|---|
| AB | 57°45' | 237°45' | N 57°45' E |
| BC | 140°00' | 320°00' | S 40°00' E |
| CD | 238°15' | 58°15' | S 58°15' W |
| DA | 323°30' | 143°30' | N 36°30' W |
Interior angles (clockwise, BB of the line coming in minus FB of the line going out): A = 143°30' − 57°45' = 85°45'; B = 237°45' − 140°00' = 97°45'; C = 320°00' − 238°15' = 81°45'; D = 58°15' − 323°30' + 360° = 94°45'. Sum = 360°00', which is (2 × 4 − 4) × 90°. The same angles come out from the uncorrected readings, which shows why the angle check cannot find local attraction on its own; you need the FB and BB comparison for that.
From bearings to latitudes, departures and plot area
Once the bearings are right, each line can be split into a north-south part (latitude) and an east-west part (departure):
- Latitude = L × cos θ, positive for N, negative for S
- Departure = L × sin θ, positive for E, negative for W
Here θ is the reduced bearing and L the measured length. Using the WCB directly in a calculator gives the same numbers with the signs worked out for you.
Worked example 3: closing error and Bowditch correction
The measured lengths of the traverse in example 2 were AB 23.20 m, BC 24.00 m, CD 24.75 m and DA 23.55 m, a perimeter of 95.50 m.
| Line | Length (m) | RB | Latitude (m) | Departure (m) | Correction lat / dep | Corrected lat / dep |
|---|---|---|---|---|---|---|
| AB | 23.20 | N 57°45' E | +12.380 | +19.621 | +0.024 / +0.002 | +12.404 / +19.622 |
| BC | 24.00 | S 40°00' E | −18.385 | +15.427 | +0.025 / +0.002 | −18.360 / +15.429 |
| CD | 24.75 | S 58°15' W | −13.024 | −21.046 | +0.025 / +0.002 | −12.998 / −21.045 |
| DA | 23.55 | N 36°30' W | +18.931 | −14.008 | +0.024 / +0.002 | +18.955 / −14.006 |
| Sum | 95.50 | −0.098 | −0.007 | 0.000 / 0.000 |
The sums of latitudes and departures should be zero for a closed figure. They are −0.098 m and −0.007 m, so the closing error is the square root of (0.098² + 0.007²) = 0.098 m. Relative closing error = 0.098 / 95.50, about 1 in 970. Accuracy needed depends on the job and on your institute or office; for a rough compass survey this is acceptable, but a cadastral boundary needs much better work with a total station.
The Bowditch rule shares the error in proportion to line length: correction to the latitude of a line = −(sum of latitudes) × (length of line ÷ perimeter), and the same for departures. For AB: 0.098 × 23.20 / 95.50 = +0.024 m. After correction both sums are zero, and the coordinates of the corners, taking A as (0, 0), are B (12.404 N, 19.622 E), C (5.957 S, 35.051 E) and D (18.955 S, 14.006 E).
The area by the coordinate (shoelace) method is 566.3 m², which is about 6,096 sq ft, or 1 ropani 1 aana 3 paisa 1 dam (1 ropani = 5,476 sq ft = 508.74 m²). You can check this kind of conversion with the Jagga Napi land area converter, and work out the area of an irregular plot from site measurements with the plot area calculator. For the unit background, see 1 ropani in square feet and square metres.
Magnetic bearing, true bearing and declination
A compass gives magnetic bearings. Maps and survey sheets use true (geographic) north. The angle between the two is the magnetic declination, and it changes with place and year. The rule:
- True bearing = magnetic bearing + declination if the declination is east
- True bearing = magnetic bearing − declination if the declination is west
Example with an assumed value: if the declination were 0°30' W, a magnetic bearing of 238°15' would be a true bearing of 237°45'. If it were 0°30' E, the true bearing would be 238°45'. Do not use these example figures for real work. Get the current declination for your location and date from the NOAA geomagnetic calculators, or use the value printed on the survey sheet you are matching.
Step-by-step method for any bearing problem
- Write every bearing in one system first. WCB is safer for working; convert RB to WCB with the table above.
- Find the back bearing of each line (FB ± 180°).
- Compare observed FB and BB. If any line is off 180°, correct local attraction starting from a line that is exactly 180°.
- Find interior angles and check the sum against (2n − 4) × 90°.
- Apply declination if the result must refer to true north.
- Convert to RB if the question asks for it, or if you are working out latitudes and departures by hand.
- Find latitudes and departures, the closing error and, if needed, adjust by the Bowditch rule.
Common mistakes seen on Nepali sites and in exams
- Writing the quadrant letters in the wrong order. RB always starts with N or S and ends with E or W. "E 58° S" is not a reduced bearing.
- Measuring RB from the wrong axis. A WCB of 238°15' is S 58°15' W, not S 31°45' W. The angle is always from the north-south line.
- Changing only one letter for a back bearing. The back bearing of N 57°45' E is S 57°45' W; both letters change.
- Setting the compass next to the rod stack. On building sites steel bars, scaffolding pipes and the mixer drum are the usual cause of local attraction. Keep keys, phones and the steel tape away from the compass too.
- Correcting local attraction from a station that is itself affected. Always start from a line whose FB and BB differ by exactly 180°.
- Mixing up minutes and decimals. 58°15' is 58.25°, not 58.15°. Convert before using a calculator, or use its degree-minute key.
- Treating a compass traverse as a legal boundary survey. Land boundaries in Nepal are fixed by the Survey Office maps and the Napi Office. A compass survey is fine for a rough plan or a site check, not for settling a boundary dispute.
The same habits of booking and checking apply in levelling; see the rise and fall method worked example and the free levelling calculator.
Frequently asked questions
What is the reduced bearing of a line with WCB 215°30'?
It is in the third quadrant, so RB = 215°30' − 180° = S 35°30' W.
What is the whole circle bearing of N 45° W?
Fourth quadrant: WCB = 360° − 45° = 315°.
Which compass reads WCB and which reads RB?
The prismatic compass has its ring graduated from 0° to 360°, so it reads WCB directly. The surveyor's compass is graduated in quadrants (0° to 90° from N and S), so it reads RB.
Does local attraction change the included angles?
No. An error at a station affects both bearings read there by the same amount, so the angle between them stays correct. That is why the FB and BB comparison is needed to detect it.
Where can I practise bearing questions?
Surveying questions on bearings, traversing and levelling come up in the civil engineering papers. Practise them on the engineering MCQ practice page and find more calculators on the free engineering tools page.
Sources and further reading
- Compass traversing lecture notes, Khulna University of Engineering and Technology: Compass Traversing (PDF)
- NOAA National Centers for Environmental Information: Magnetic field and declination calculators
- Survey Department, Government of Nepal: dos.gov.np (official maps and cadastral survey)
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