How to Calculate Whole Circle Bearings and Distances from Co-Ordinates.

Why do we need to calculate the Whole Circle Bearing and Distance between two sets of Co-Ordinates?

Information is supplied in all manner of forms, and quite often it is not in the form that we need to use it in.  We need to transform this information into a format we can use.  For many years, Site Engineers and Surveyors would use Theodolites and tape measures or Total Stations that only provided information in a format of bearings and distances. 

Predominately though most surveyors and engineers will be interested in physically checking a distance between two features with a tape measure.  The theoretical positions of these features will be given in two sets of co-ordinates (in the format of eastings and northings).  The engineer or surveyor will be wanting to know what the theoretical distance is compared to the actual distance measured between the features.

What is a Whole Circle Bearing?

The term Whole Circle Bearing defines the angle (or direction) of a survey line from True North in a clockwise direction. 

The Whole Circle Bearing is the total angle from North that we need to turn to aim towards the second set of co-ordinates being considered.  This is measured in a clockwise direction from North.

How does a Site Engineer Calculate bearings and distances from easting and northings?

A site engineer will often be given information for setting out a site in the form of co-ordinates and it will be their job to ensure that these points are accurately marked out on site.  For this setting out, the site engineer would have either a theodolite and tape measure or a total station.  The latest total stations will take care of all these calculations, but it is always handy to know how this is done for checking purposes and for the site engineers less fortunate to not have the latest total stations to hand.

To show how to do these calculations I am going to use some real-world data so that you can see the sort of information a site engineer would use and the format it will be presented in.  The following data is the standard cross-sectional data that would be given for a typical road construction project in the UK. 

In this example calculation we shall calculate the Whole Circle Bearing and Distance between points M001 and CL01.

Cross-sectional information should look something like this: –

CHAINAGE      40.000

            OFFSET         R.L.           LABEL CUT               EASTING                 NORTHING

    1      -10.010      136.931          V001                   394499.950            295852.514

    2       -8.955       136.934          F001                    394499.177            295853.232

    3       -5.941       136.723          CL01                    394496.968            295855.282

    4        0.000       136.787          M001                  394492.613            295859.323

    5        6.876       136.861          CR01                    394487.573            295864.000

    6        9.583       137.072          V002                    394485.589            295865.841

When working with cross-sectional information in relation to roads the convention adopted is that we always look up chainage (increasing chainage) and that we consider the Left Hand Side to be a negative offset from the centreline and the Right Hand Side to be a positive offset from the centreline.

How to calculate the distance between two co-ordinated points?

As site engineers will receive the co-ordinated information in an easting and northing format, it will be straightforward to work out the plan distance between the two co-ordinated points.  In most cases the plan distance will be just as accurate as using the slope distance.  If the slope distance is going to be of importance then a further calculation, using the same method, would be required.

To calculate the distance between two co-ordinated points we need to use the Pythagoras Theorem.

a² + b² = c²

(394496.968-394492.613)² + ( 295855.282-295859.323)²=c²

4.355² + (-4.041)² = c²

18.966 + 16.330 = c²

(Square Root) 35.296 = c

5.941 = c

As you can see this is the same dimension as the OFFSET distance from M001 to CL01 in the table above.

How does a Site Engineer Calculate the Whole Circle Bearing Between Two Points?

Now we need to work out the WCB (Whole Circle Bearing) from M001 to CL01.  To calculate the WCB we need to use trigonometry.

What is the Whole Circle Bearing?

Plotting the relative co-ordinated points will help identify the WCB value.  The Whole Circle Bearing is the total angle from North that is the direction of travel from the points being considered.  This is measured in a clockwise direction from North.

From the information already calculated above, we have the lengths of all three sides of the right angled triangle to use for working out the bearing.  That is the Hypotenuse (5.491m), difference in Eastings (4.355m) and the difference in Northings (4.041). 

So, using trigonometry, we can work out the value for the angle x for a right-angled triangle.

SOHCAHTOA

For this example it is best to use the Tan Function.  So, using the Tan function, we get

Tan (x) = Opposite / Adjacent

Tan (x) = -4.041 / 4.355                 

x= 42 Degrees 51 Minutes 30 Seconds

Whole Circle Bearing Diagram
M001 to CL01 Survey Line

As we have drawn out the direction between M001 and CL01 above, we know that we now need to add 90 Degrees to the angle x we have calculated to give a WCB of 132 Degrees 51 Minutes 30 Seconds. This is because the Whole Circle Bearing is measured from True North.

What are the Calculated Results Between the Two Points?

The Whole Circle Bearing between M001 and CL01 is 132 Degrees 51 Minutes 30 Seconds.

The Distance between M001 and CL01 is 5.941m.


Spreadsheet for Calculating the Distance and Whole Circle Bearings between two sets of co-ordinates.

There is a free spreadsheet available on the Lichfield Survey Supplies website for calculating the whole circle bearing and distance between two sets of co-ordinates.  You can get this spreadsheet by following this link to the LSSL Inverse Calculator Spreadsheet.