How to Calculate Whole Circle Bearings & Distances from Co-Ordinates

In surveying and construction, information is rarely handed to you in the exact format you need. Most projects provide the locations of features, boundaries, or design points as sets of coordinates—Eastings and Northings—rather than as direct distances and bearings. Yet on site, when you need to set out a fence line, verify as-built positions, or check progress, what you really need to know is how far apart two points are and in which direction you need to travel between them.

This is where understanding how to calculate whole circle bearings and distances from co-ordinates becomes an essential skill for every engineer and surveyor. By converting coordinate data into meaningful distances and bearings, you bridge the gap between office-based design and practical site work. Whether you’re double-checking a layout with a tape, comparing instrument results, or troubleshooting discrepancies, knowing how to perform these calculations gives you the confidence to trust your measurements.

In this article, we’ll break down the step-by-step method for working out whole circle bearings and distances from any two sets of co-ordinates. We’ll use real-world data, clear formulas, and worked examples—plus tips, downloadable tools, and common pitfalls to avoid—so you can apply these calculations quickly and accurately on any site.

This method is exactly how I was first taught to work out whole circle bearings and distances, and it remains a core part of my daily workflow on site. Over the years, I’ve found that being able to quickly and confidently convert co-ordinates into distances and bearings is invaluable—not just for set-out and checking, but for troubleshooting and cross-checking measurements under pressure. To make my working life more efficient, I developed a simple spreadsheet that automates these calculations. It’s freely available on the Lichfield Survey Supplies website, and I use it almost every day to save time and reduce errors, whether I’m planning a job or solving a problem on site.

What is a Whole Circle Bearing?

A Whole Circle Bearing (WCB) is the clockwise angle measured from true north.

In practice, this means if you stand at your starting point and face north, the whole circle bearing tells you exactly how many degrees you need to turn to point directly at your destination point. Bearings are always measured in a full circle—so a bearing of 0° means you’re facing due north, 90° is due east, 180° is due south, and 270° is due west. By always measuring clockwise from north, whole circle bearings avoid the ambiguity of quadrant bearings and make it easy to communicate directions unambiguously, whether you’re working with a compass, a theodolite, or digital survey equipment. This system is essential for accurately setting out lines, navigating between points, or converting coordinate data into practical field instructions.

Why Calculate Bearings and Distances from Coordinates?

Most modern site layouts and survey data are based on coordinates. But on the ground, engineers need to know how far to measure (distance) and in what direction (bearing) between two features, pegs, or stations. Even with modern total stations and GNSS equipment that can perform these calculations automatically, understanding the manual process allows you to:

Check results and troubleshoot errors
Set out points with basic equipment
Validate “as built” measurements
Understand survey reports or designs
Work confidently with older instruments

Step-by-Step: Calculating Distance and Whole Circle Bearing

1. Gather Your Coordinates

Let’s use this example from a typical UK road cross-section:

No.	Offset	R.L.	Label	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

We’ll calculate the distance and bearing from M001 (start) to CL01 (finish).

This information is from a road cross-section report. 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.

2. Calculate the Differences (ΔE and ΔN)

ΔE = Easting (CL01) – Easting (M001) = 394496.968 – 394492.613 = 4.355
ΔN = Northing (CL01) – Northing (M001) = 295855.282 – 295859.323 = -4.041

To work out the distance between two points with known coordinates, we use Pythagoras’ theorem—something most of us were taught at school and, at the time, probably wondered when we’d ever need it! In surveying, it turns out to be incredibly useful. Since the difference in Eastings and Northings forms the two shorter sides of a right-angled triangle, the direct distance between the points (the “hypotenuse”) can be found using the familiar formula:
a² + b² = c²

By plugging in the differences in Eastings and Northings for “a” and “b,” you get the straight-line, plan distance “c” between the two points. It’s a perfect example of how a classroom formula finds a real-world purpose on site.

3. Calculate the Plan Distance

The distance is simply the hypotenuse of the right triangle formed by ΔE and ΔN:

Formula:
Distance = √[(ΔE)² + (ΔN)²] (Rearranged Pythagoras Formula)

Calculation:
= √[(4.355)² + (-4.041)²]
= √[18.966 + 16.330]
= √[35.296]
= 5.941 m

4. Calculate the Bearing

Next, let’s determine the Whole Circle Bearing (WCB) from M001 to CL01. Calculating the bearing between two coordinate points requires a bit of trigonometry, but it’s straightforward once you know the process. By considering the relative positions of the two points, we can use their coordinate differences to precisely find the angle—or direction—you’d need to travel from one point to the other.

To calculate the bearing, we use the tangent (Tan) function from trigonometry, based on SOHCAHTOA. In the context of bearings from north:

The difference in Northings (ΔN) is the “opposite” side (since north is your reference direction).
The difference in Eastings (ΔE) is the “adjacent” side.

So, the formula becomes:

Tan(θ) = Opposite / Adjacent = ΔN / ΔE

For our example:

Tan(θ) = -4.041 / 4.355

This gives:

θ = arctan(-4.041 / 4.355) ≈ -42° 51′ 30″

This is the angle from the 90° line heading East. We need to do a further calculation to have the Whole Circle Bearing (WCB) worked out. In our case we simply need to add 90° to our calculated angle.

Whole Circle Bearing (WCB) = 90° + arctan (ΔN / ΔE)

So:

= 90° + (42° 51′ 30″)
= 132° 51′ 30″

5. Quick Reference Table

Step	Formula	Example Result
ΔE	Easting₂ – Easting₁	4.355
ΔN	Northing₂ – Northing₁	-4.041
Distance	√[(ΔE)² + (ΔN)²]	5.941 m
Whole Circle Bearing (WCB)	90° + arctan (ΔN / ΔE)	132° 51′ 30″

Example Application: Why It Matters

Suppose you’re setting out features on a site with just a tape and a compass, or you want to check the alignment of kerbs or fencing between design coordinates. Calculating the distance and WCB lets you physically mark out or verify work on the ground—even without advanced instruments. It’s also crucial for validating “as-built” surveys, resolving discrepancies, and checking site progress.

Common Errors and Tips

Use consistent units: Make sure your Eastings and Northings are both in metres (not mm or ft).
Watch for sign errors: If the ΔN or ΔE is negative, it affects the bearing quadrant.
Double-check calculator functions: Use atan2 if available to get the correct bearing direction.
Convert decimals to degrees-minutes-seconds: Many calculators give angles in decimals; convert to DMS for bearings.
Check your result: Bearings should be between 0° and 360°. If you get a negative value, add 360°.

Downloadable Tool

Speed up your work with the free LSSL Inverse Calculator Spreadsheet, designed to calculate bearings and distances from coordinate data automatically. Download it and streamline your site calculations.

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Summary

Calculating whole circle bearings and distances from coordinates is a practical skill for every site engineer and surveyor. With a simple understanding of coordinate differences, the Pythagoras theorem, and bearing formulas, you can bridge the gap between design data and real-world setting out. Whether you use the calculations as a cross-check or your main method, these steps will give you confidence and accuracy on site.