SW: Az = 180° + α · NW: Az = 360° − α
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NE:
Az = αSE:
Az = 180 − αSW:
Az = 180 + αNW:
Az = 360 − α
0–90:
N α° E90–180:
S (180−Az)° E180–270:
S (Az−180)° W270–360:
N (360−Az)° W
Back = (Az + 180) mod 360
Az = atan2(ΔE, ΔN) × 180/πNote: surveying uses
atan2(ΔE, ΔN) not atan2(ΔN, ΔE)
True = Magnetic + DeclinationEast = + / West = −
Bearing to Azimuth Conversion — Complete Engineering Guide
Converting bearing to azimuth is a daily task for land surveyors, civil engineers, pilots, and GIS professionals. The bearing system uses four quadrant notations (NE, SE, SW, NW) with angles from 0° to 90°. Azimuth is a continuous 0° to 360° measurement clockwise from north. This free online bearing to azimuth calculator handles all formats, including DMS (degrees, minutes, seconds), decimal degrees, and radians, with a live rotating compass.
Bearing to Azimuth Formula
The conversion depends entirely on which quadrant the bearing falls in. Apply the correct formula using the angle within the quadrant (α, always 0° to 90°):
| Quadrant | Bearing Example | Formula | Azimuth Result |
|---|---|---|---|
| NE | N 45° E | Azimuth = α | 45° |
| SE | S 45° E | Azimuth = 180° − α | 135° |
| SW | S 45° W | Azimuth = 180° + α | 225° |
| NW | N 45° W | Azimuth = 360° − α | 315° |
All results are normalized to the 0°–360° range. Back azimuth is calculated automatically as (Azimuth + 180°) mod 360°.
How to Convert Bearing to Azimuth — Step by Step
Step 1: Read the quadrant from the bearing. "S 36° E" is the SE quadrant, angle = 36°.
Step 2: Apply the SE formula. Azimuth = 180° − 36° = 144°.
Step 3: Back azimuth = (144° + 180°) mod 360° = 324°.
The tool above accepts free-text bearing input in any common format: N30E, S 45° 30' W, N 45.5 E, or the manual DMS builder in the left panel.
Azimuth to Bearing Conversion
Surveyors often need the reverse: converting a total station or GPS azimuth back to quadrant bearing notation for use in legal property descriptions. Switch to the "Az → Bearing" tab and enter any azimuth from 0° to 360°. The tool outputs the full quadrant bearing in both decimal and DMS format. For example, 225.5° converts to S 45° 30' W.
Batch Bearing to Azimuth Converter
The Batch mode lets you paste a column of bearings — one per line — and converts all of them at once into a table showing azimuth, DMS, bearing, back azimuth, and radians. This is a major time saver for surveyors entering traverse data or GIS analysts processing boundary descriptions. Export the full table to CSV with one click.
Back Azimuth Calculator
The back azimuth is the reciprocal direction: add 180° to the forward azimuth and apply modulo 360°. A forward azimuth of 127.458° gives a back azimuth of 307.458°. Surveyors use back azimuths to verify traverse closure — if the computed back azimuth from the endpoint matches the measured back sight, the measurement is confirmed.
Magnetic Declination Correction
Compass instruments read magnetic bearings, which differ from true (geodetic) bearings by the magnetic declination at your location. The correction is: True Bearing = Magnetic Bearing + Declination. East declination is positive, west is negative. In Seattle, declination is approximately +15.5° East, so a compass reading of 90° gives a true bearing of 105.5°. This tool includes preset declination values and shows both needles on the compass — orange for magnetic, green for true.
Output Format Options
Switch between three output formats using the buttons in the left panel footer. Decimal Degrees (144.0000°) is standard for most software. DMS (144° 00' 00.0") matches traditional surveying documentation. Radians (2.5133) is needed for GIS calculations and trigonometry. Grad (gon) values are also shown in the Info tab, used in European surveying systems.
Who Uses Bearing and Azimuth Conversions?
Land surveyors convert azimuth output from total stations to quadrant bearing notation for property deed descriptions, which legally require the N/S format in most jurisdictions.
Civil engineers use azimuths for highway horizontal alignment, pipeline routing, and construction staking. Traverse calculations accumulate azimuth through each course.
Pilots apply magnetic declination to convert compass readings to true course before filing flight plans. A 15° error over a two-hour flight at 140 knots misses the destination by 72 nautical miles.
GIS analysts need radians and atan2-based azimuth from coordinate pairs for spatial analysis and feature orientation calculations.