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Whole Circle Bearing to Reduced Bearing (WCB to RB) | Surveying

Updated: Jan 19


Whole Circle Bearing to Reduced Bearing Conversion in surveying
Conversion of WCB to RB example

In Whole Circle Bearing - Bearings are measured clockwise from north of reference meridian; In Reduced Bearing - Bearings are measured either clockwise or anti-clockwise from north/south whichever is close to the line.


Let us see a few basic concepts before we jump on the conversion steps. If you already know them, skip directly to the WCB to RB conversion steps


What is bearing?

Bearing is the direction of survey lines with respect to a meridian

Bearing is as simple as the direction of survey lines w.r.t. a meridian.


Meridian is a fixed reference line.


The bearing system is of two types - Whole circle bearing and reduced bearing. We can convert from one system to another.


Whole Circle Bearing (WCB)


Bearings are measured clockwise from north of reference meridian.


It is also known as the Azimuthal System

  • It varies from 0 degrees to 360 degrees in the clockwise direction

  • North will be zero degrees; East will be 90 degrees; South will be 180 degrees and West will be 270 degrees.

  • Prismatic compass uses the WCB system

Examples of WCB


Line from origin to A (OA) : The angle is measured from the north to OA represented by θ in the figure.


Line from origin to B (OB): The angle is measured from the north to OB represented by ϕ in the figure.


Whole Circle Bearing (WCB)
Whole Circle Bearing (WCB)

Reduced Bearing (RB)


Bearings are measured either clockwise/anti-clockwise from north/south whichever is close to the line.

It is also known as Quadrantal Bearing (QB)

  • Angles vary from 0 degrees to 90 degrees

  • It is measured from either clockwise/anti-clockwise from north/south (If the line is closer to south and going towards west, it is read as Sβ°W)

  • Four quadrants are possible namely NE, SE, SW and NW

  • Surveyor's compass used RB system

Examples of RB


Line from origin to A (OA) : The angle is measured from the north to OA because OA is closer to the north. It is represented by NαE in the figure.


Line from origin to B (OB): The angle is measured from the south to OB because OB is closed to the south. It is represented by SβW in the figure.


Reduced Bearing (RB)
Reduced Bearing (RB)

Conversion from Whole Circle Bearing to Reduced Bearing


It is very easy to convert if we draw diagrams as shown below.

  1. In this example, WCB = 240°

  2. It lies in the third quadrant as the angle is between 180° and 270°

  3. So, the quadrant position reading as per RB is SW

  4. The angle will be 240° - 180° = 60°

  5. RB = S60°W

There are pin-pointed formulas to convert WCB to RB. But, diagrams make it much easier.

Conversion from Whole Circle Bearing to Reduced Bearing examples
Example: Convert WCB 240° to RB

Challenge yourself with a question on WCB to RB


3 Simple Things to Remember: WCB and RB


Download 3 simple things:

  • What is WCB?

  • What is RB?

  • Convert WCB to RB using diagrams

Whole Circle Bearing and Reduced Bearing
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Formula to convert WCB to RB

  1. When WCB lies between 0° and 90°: RB = WCB

  2. When WCB lies between 90° and 180 degrees: RB = 180 - WCB

  3. When WCB lies between 180° and 270 degrees: RB = WCB - 180

  4. When WCB lies between 270° and 360 degrees: RB = 360 - WCB

In the above formula, based on the quadrants use NE, SE, SW and NW accordingly.


Example if WCB is 240 degrees, it lies in the third quadrant. Use SW.

  • RB = WCB - 180° = 240° - 180° = 60°

  • RB = S60W

Watch the full video on whole circle bearing and reduced bearing here

To convert RB to WCB, you can use the same formulas. You can as well draw the diagram and check if it is all too confusing.


More Examples on WCB to RB


Convert WCB = 35° to RB

  • It is in the First quadrant (NE)

  • Angle = 35°

  • RB = N35°E

Convert WCB = 130° to RB

  • It is in the Second quadrant (SE)

  • Angle = 180 - 130° = 50°

  • RB = S50°E

Convert WCB = 240° to RB

  • It is in the Third quadrant (SW)

  • Angle = 240° - 180° = 60°

  • RB = S60°W

Convert WCB = 290° to RB

  • It is in the Fourth quadrant (NW)

  • Angle = 35°

  • RB = N70°W

Challenge Question


Start the quiz below. Solution included.


Now, try converting S89°W to WCB​ in your mind. Comment your answer.


I hope you have become an expert and found what you were looking for. If you still have any questions, drop a comment below.


These conversions become simpler if you understand the basic definition for WCB and RB, and if you can draw these diagrams.


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