Bearing to Azimuth Converter (Quadrant Bearings ↔ Azimuths, DMS)
Convert a surveying quadrant bearing such as N 45°30′ E or S 20° W into an azimuth, or an azimuth back into a bearing — in degrees, minutes and seconds, with back bearing and back azimuth.
How it works
Bearing → azimuth: N θ E = θ · S θ E = 180° − θ · S θ W = 180° + θ · N θ W = 360° − θ · azimuth → bearing reverses these by quadrant · back azimuth = azimuth ± 180° · back bearing swaps N↔S and E↔W with the same angle · decimal degrees = D + M/60 + S/3600.
Surveyors and deed descriptions give directions two ways. A quadrant bearing starts from north or south and turns up to 90° towards east or west (N 45°30′ E); an azimuth is a single angle from 0° to 360° measured clockwise from north (45°30′). Azimuths are easier to compute with — latitudes and departures, coordinate geometry — while bearings remain common in plats and legal descriptions. The back bearing of a line is its direction travelled the other way: the same angle with both letters swapped.
Worked example
The bearing N 45°30′00″ E lies in the first (north-east) quadrant, so its azimuth equals the angle: 45°30′00″, or 45.5000° in decimal degrees. The back azimuth is 225°30′00″ and the back bearing is S 45°30′00″ W. Going the other way, an azimuth of 225.75° is S 45°45′00″ W.
Sources: Azimuth — surveying azimuths and bearings · Bearing (angle) — quadrant bearings in land navigation and surveying
Frequently asked questions
What is the difference between a bearing and an azimuth?
An azimuth is measured clockwise from north through a full 360°. A quadrant bearing is an angle of at most 90° measured from north or south towards east or west, written like N 30° E or S 60° W.
How do I convert S 30° E to an azimuth?
In the south-east quadrant subtract the angle from 180°: 180° − 30° = 150°. For S 30° W add it to 180° (210°) and for N 30° W subtract it from 360° (330°).
What is a back bearing?
The bearing of the same line measured from its other end. Keep the angle and swap both letters: the back bearing of N 45° E is S 45° W. In azimuths, add or subtract 180°.
Are these true, grid or magnetic directions?
The conversion is the same for all three. To change between magnetic and true north, apply the magnetic declination — the magnetic declination calculator does that step.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.