CADGISHow to use the calculator
- Enter the X (Easting) and Y (Northing) of Point A, where the line starts, and Point B, where it ends. Names are optional.
- Press Calculate or the Enter key. After that, results update as you edit.
- Choose the decimal places for distance, azimuth and bearing seconds.
- Use Copy results to paste everything into a report or field book.
Distance uses the same unit as your coordinates, for example meters for UTM. Coordinates must be on a flat grid (projected). Latitude and longitude in degrees won't give correct results.
Formulas
With X = Easting and Y = Northing:
ΔX = XB − XA
ΔY = YB − YA
Distance = √(ΔX² + ΔY²)
Azimuth = atan2(ΔX, ΔY), normalised to 0° ≤ Az < 360°
Back azimuth = (Azimuth + 180°) mod 360°
Surveying azimuths are measured clockwise from North. Mathematics measures angles counter-clockwise from the X axis (East). That's why ΔX comes first in atan2(ΔX, ΔY). Swapping them to atan2(ΔY, ΔX) is a common mistake and gives a wrong direction.
Azimuth to quadrant bearing
A quadrant bearing is an acute angle (0°–90°) measured from North or South toward East or West. The signs of ΔX and ΔY decide the quadrant:
| ΔY | ΔX | Azimuth range | Quadrant bearing |
|---|---|---|---|
| + | + | 0° – 90° | N (Az) E |
| − | + | 90° – 180° | S (180° − Az) E |
| − | − | 180° – 270° | S (Az − 180°) W |
| + | − | 270° – 360° | N (360° − Az) W |
When ΔX = 0 the line is due North or due South. When ΔY = 0 it's due East or due West.
Worked examples
| A (X, Y) | B (X, Y) | ΔX | ΔY | Distance | Azimuth | Bearing |
|---|---|---|---|---|---|---|
| 500000, 2800000 | 500100, 2800100 | 100 | 100 | 141.421 | 45.0000° | N 45° 00' 00.00" E |
| 1000, 1000 | 1030, 960 | 30 | −40 | 50.000 | 143.1301° | S 36° 52' 11.63" E |
| 1000, 1000 | 960, 970 | −40 | −30 | 50.000 | 233.1301° | S 53° 07' 48.37" W |
| 1000, 1000 | 988, 1005 | −12 | 5 | 13.000 | 292.6199° | N 67° 22' 48.49" W |
Example: southeast line step by step
A = (1000, 1000), B = (1030, 960). ΔX = 30 and ΔY = −40, so the distance is √(30² + 40²) = 50.000. The azimuth is atan2(30, −40) = 143.1301° (143° 07' 48.37"). ΔY is negative and ΔX is positive, so the line points southeast: 180° − 143.1301° = 36.8699°, giving S 36° 52' 11.63" E. The back azimuth is 323.1301°, which is N 36° 52' 11.63" W.
Frequently asked questions
What is the difference between azimuth and bearing?
An azimuth is a single angle from 0° to 360°, measured clockwise from North. A quadrant bearing gives the same direction as an angle up to 90° from North or South, plus E or W. For example, azimuth 225° is the same as S 45° W.
What is a back bearing?
It's the direction from Point B back to Point A. The back azimuth differs by exactly 180°. For the quadrant bearing, the letters swap: N ↔ S and E ↔ W.
Is this true north or grid north?
The result is relative to the north of your coordinate grid, so with UTM coordinates it's grid north. Converting to true north requires the grid convergence angle for your location.
Can I use negative or large UTM coordinates?
Yes. Local grids with negative values and full UTM coordinates such as 500000 / 2800000 both work.