The same direction can be written down in two completely different ways. “One hundred and thirty five degrees” and “South 45 degrees East” describe the identical line — they are just two conventions for saying the same thing.
This post covers both systems, and the conversion between them that examinations test constantly.
The Two Systems
- Whole circle bearing system (W.C.B.), also called the azimuthal system
- Quadrantal bearing system (Q.B.)
1. Whole Circle Bearing (W.C.B.)
In this system, the bearing of a line is measured from the north in a clockwise direction. The value varies from 0° to 360°.
The Three Rules
| Feature | Rule |
|---|---|
| Starting point | Always north |
| Direction of measurement | Always clockwise |
| Range | 0° to 360° |
Everything about W.C.B. is fixed. There is no choice to make, which is exactly why the system is convenient for calculation — you can add and subtract bearings directly without worrying about signs or quadrants.
The prismatic compass is graduated in the whole circle bearing system.
2. Quadrantal Bearing (Q.B.)
In this system, the bearing of a line is measured eastward or westward from north or south, whichever is nearer. The value varies from 0° to 90°.
How a Quadrantal Bearing Is Written
A quadrantal bearing always has three parts:
- A letter — N or S — the meridian it is measured from
- An angle between 0° and 90°
- A letter — E or W — the direction of turning
So N 30° E means: start at north, turn 30 degrees towards the east.
The “Whichever Is Nearer” Rule
This phrase is what keeps the angle below 90 degrees.
Consider a line pointing almost due south. Measuring from north would give an angle near 180 degrees — outside the permitted range. Measuring from south gives a small angle instead. So the rule is to always reference the nearer of the two poles, which guarantees the result never exceeds 90 degrees.
The surveyor’s compass is graduated in the quadrantal bearing system.
Reduced Bearing (R.B.)
When the whole circle bearing of a line is converted into the quadrantal system, it is termed the reduced bearing.
So “reduced bearing” and “quadrantal bearing” describe the same thing — the R.B. is simply the Q.B. obtained by converting from a W.C.B.
The word reduced makes sense here: a value that could be anything up to 360 degrees is reduced to one no greater than 90.
The Four Quadrants
Conversion depends on which quadrant the line falls in, so identify that first.
| Quadrant | W.C.B. Range | Quadrant Name |
|---|---|---|
| I | 0° to 90° | NE |
| II | 90° to 180° | SE |
| III | 180° to 270° | SW |
| IV | 270° to 360° | NW |
The Conversion Table — W.C.B. to R.B.
| W.C.B. Between | Rule | Quadrant |
|---|---|---|
| 0° and 90° | R.B. = W.C.B. | N … E |
| 90° and 180° | R.B. = 180° − W.C.B. | S … E |
| 180° and 270° | R.B. = W.C.B. − 180° | S … W |
| 270° and 360° | R.B. = 360° − W.C.B. | N … W |
How to Remember It Without Memorising
The four rules look arbitrary until you see the pattern behind them. Every rule is doing the same thing — finding the angle from the nearer of north or south.
- Quadrant I (0–90°): already measured from north, already under 90°. Nothing to do.
- Quadrant II (90–180°): south is at 180°, so the angle back to south is 180 − W.C.B.
- Quadrant III (180–270°): south is at 180° and we have gone past it, so the angle from south is W.C.B. − 180
- Quadrant IV (270–360°): north is at 360°, so the angle back to north is 360 − W.C.B.
In every case you are simply asking: how far is this line from the nearest of north (0° or 360°) or south (180°)?
Getting the Letters Right
The two letters follow the same logic:
- The first letter is whichever of N or S you measured from — N for quadrants I and IV, S for quadrants II and III.
- The second letter is the side you turned towards — E for quadrants I and II (the eastern half, 0° to 180°), W for quadrants III and IV (the western half, 180° to 360°).
Worked Examples
| W.C.B. | Quadrant | Working | R.B. |
|---|---|---|---|
| 45° | I (NE) | Same as W.C.B. | N 45° E |
| 135° | II (SE) | 180 − 135 = 45 | S 45° E |
| 210° | III (SW) | 210 − 180 = 30 | S 30° W |
| 300° | IV (NW) | 360 − 300 = 60 | N 60° W |
Converting Back — R.B. to W.C.B.
Simply reverse each rule:
| R.B. Quadrant | W.C.B. = |
|---|---|
| N … E | R.B. |
| S … E | 180° − R.B. |
| S … W | 180° + R.B. |
| N … W | 360° − R.B. |
The Two Systems Compared
| Whole Circle Bearing | Quadrantal Bearing | |
|---|---|---|
| Measured from | North only | North or south, whichever is nearer |
| Direction | Clockwise only | East or west |
| Range | 0° to 360° | 0° to 90° |
| Written as | A single number | Letter, angle, letter |
| Instrument | Prismatic compass | Surveyor’s compass |
| Also called | Azimuthal system | Reduced bearing when converted from W.C.B. |
Which Is Better?
Neither — they serve different purposes.
W.C.B. is better for computation. A single number from 0 to 360 can be added and subtracted directly, which is why traverse calculations use it.
Q.B. is better for visualisation. “S 45° E” instantly tells you the line runs south-east; “135°” requires a moment’s thought. It is also the natural form for trigonometry, since the angle is always acute.
Quick Revision Notes
- Two systems: whole circle bearing (azimuthal) and quadrantal bearing.
- W.C.B.: measured from north, clockwise, range 0° to 360°. Used on the prismatic compass.
- Q.B.: measured east or west from north or south, whichever is nearer, range 0° to 90°. Used on the surveyor’s compass.
- Reduced bearing = a W.C.B. converted into the quadrantal system.
- Conversion: 0–90° → R.B. = W.C.B. (NE); 90–180° → 180 − W.C.B. (SE); 180–270° → W.C.B. − 180 (SW); 270–360° → 360 − W.C.B. (NW).
- First letter is N for quadrants I and IV, S for II and III. Second letter is E for I and II, W for III and IV.
Mistakes Students Commonly Make
- Confusing the two subtraction rules for the southern quadrants. Quadrant II is 180 − W.C.B.; quadrant III is W.C.B. − 180.
- Getting the quadrant letters the wrong way round. For 135° the answer is S 45° E, not N 45° E.
- Giving a quadrantal bearing greater than 90°. By definition it cannot exceed 90°.
- Omitting one or both letters. A quadrantal bearing needs three parts to be complete.
- Swapping the instruments. Prismatic compass reads W.C.B.; surveyor’s compass reads Q.B.
- Measuring W.C.B. anticlockwise. It is always clockwise from north.
- Thinking reduced bearing is a third system. It is simply the quadrantal bearing obtained by conversion.
Conclusion
Two ways of writing the same direction. The whole circle system uses one number from 0 to 360, always clockwise from north, which makes it ideal for calculation and is what the prismatic compass shows. The quadrantal system uses a letter, an acute angle and another letter, referencing whichever pole is nearer, which makes it easy to picture and is what the surveyor’s compass shows. Converting between them comes down to a single question — how far is this line from the nearer of north or south — and the four rules follow from that automatically.
Frequently Asked Questions
What are the two systems of designating bearings?
The whole circle bearing system, also called the azimuthal system, and the quadrantal bearing system.
What is a whole circle bearing?
A bearing measured from the north in a clockwise direction, with values ranging from 0 to 360 degrees.
What is a quadrantal bearing?
A bearing measured eastward or westward from north or south, whichever is nearer, with values ranging from 0 to 90 degrees.
What is a reduced bearing?
The value obtained when a whole circle bearing is converted into the quadrantal system.
How do you convert a W.C.B. between 90 and 180 degrees?
Reduced bearing = 180° − W.C.B., and the quadrant is S … E.
How do you convert a W.C.B. between 180 and 270 degrees?
Reduced bearing = W.C.B. − 180°, and the quadrant is S … W.
How do you convert a W.C.B. between 270 and 360 degrees?
Reduced bearing = 360° − W.C.B., and the quadrant is N … W.
What is the reduced bearing of a line whose W.C.B. is 210 degrees?
Since 210 lies between 180 and 270, R.B. = 210 − 180 = 30, giving S 30° W.
Which compass uses which system?
The prismatic compass is graduated in the whole circle bearing system, while the surveyor’s compass is graduated in the quadrantal bearing system.
Why does a quadrantal bearing never exceed 90 degrees?
Because it is always measured from whichever of north or south is nearer to the line, so the angle can never be more than a quarter turn.