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Surveying

Azimuths: Grid, True and Magnetic North

Updated · GeoCAD Bridge engineering team · 8 min read

Summary

An azimuth is a horizontal direction measured clockwise from north, from 0° to 360°. In surveying it is computed from coordinates as atan2(ΔE, ΔN). Grid azimuths on a map projection differ from true (geodetic) azimuths by the grid convergence angle, and from magnetic azimuths by convergence plus magnetic declination.

Definition

Azimuth: the clockwise horizontal angle from a reference north direction to a line, between 0° and 360°.

  • Az = atan2(ΔE, ΔN)
  • Back azimuth = Az ± 180°
  • True = Grid + γ (sign convention!)
  • UTM γ ≈ Δλ·sin φ

Azimuth from coordinates (inverse)

ΔE = E2 − E1, ΔN = N2 − N1 Az = atan2(ΔE, ΔN) (add 360° if negative) Distance = √(ΔE² + ΔN²)

Note the argument order: easting first, because azimuths are measured from north. Using atan2(ΔN, ΔE) gives a mathematical angle from east, a common bug.

Grid, true and magnetic north

Grid convergence (γ) is the angle between grid north and true north at a point. On Transverse Mercator and UTM grids it is approximately γ ≈ Δλ·sin φ, where Δλ is the longitude difference from the central meridian. GeoCAD Bridge reports γ with each transformed point; it is positive east of the central meridian in the northern hemisphere, meaning grid north lies east of true north and true azimuth = grid azimuth + γ.

Magnetic declination is the angle between true and magnetic north and changes over time; obtain it from a current geomagnetic model such as the WMM.

Worked example

From P1 (500 000.000 E, 2 700 000.000 N) to P2 (500 250.000 E, 2 699 900.000 N):

ΔE = +250.000, ΔN = −100.000 Az = atan2(250, −100) = 111.8014° = 111°48′05″ Distance = 269.258 m

FAQ

Why does my azimuth come out 90° wrong?

The arguments to atan2 are probably swapped. Use atan2(ΔE, ΔN) for azimuths from north.

Is grid convergence the same everywhere in a UTM zone?

No. It is zero on the central meridian and grows with distance from it and with latitude, reaching about ±3° at the zone edges at mid-latitudes.

Tools

Articles

References

  1. Ghilani, C. D. & Wolf, P. R. Elementary Surveying: An Introduction to Geomatics, 15th ed., Pearson, 2018 (traverse computations and adjustments).
  2. Snyder, J. P. Map Projections: A Working Manual, USGS Professional Paper 1395, 1987.
  3. NOAA NCEI. World Magnetic Model.

Cite this page

GeoCAD Bridge. “Azimuths: Grid, True and Magnetic North.” GeoCAD Bridge Knowledge Center, updated 2026-10-05. https://www.geocadbridge.com/knowledge/survey-azimuths/