Azimuth Calculator: Bearing from Coordinates, Back-Azimuth and Conversion
Enter two coordinate pairs to get the forward azimuth (true bearing clockwise from North), the back-azimuth, the compass bearing, the quadrant notation, and the great-circle distance. Switch to Convert mode to translate any azimuth to a compass bearing or quadrant notation without needing coordinates. Every result comes with a show-your-work step panel.
What is azimuth?
Azimuth is the horizontal angle measured clockwise from true North to a target direction, expressed in degrees from 0 to 360. An azimuth of 0 deg (or 360 deg) points due North, 90 deg points due East, 180 deg points due South, and 270 deg points due West. It is the standard directional reference used in navigation, surveying, astronomy, and military operations because it gives an unambiguous single number for any direction - unlike the quadrant bearing notation (e.g. N45E) that divides the compass into four quadrants and requires specifying the quadrant letter as well as the angle.
How the azimuth formula works
When you provide two geographic coordinates, this calculator uses the spherical trigonometry forward-azimuth formula:
- Convert both latitudes (phi1, phi2) and the longitude difference (delta-lambda) to radians.
- Compute two components: Y = sin(delta-lambda) x cos(phi2) and X = cos(phi1) x sin(phi2) - sin(phi1) x cos(phi2) x cos(delta-lambda).
- The initial bearing in radians is theta = atan2(Y, X).
- Convert to degrees and normalize to 0-360: azimuth = ((theta x 180/pi) % 360 + 360) % 360.
The great-circle distance is calculated with the Haversine formula using an Earth radius of 6,371 km. Both formulas assume a spherical Earth, which introduces a maximum error of about 0.3% compared to the more precise Vincenty formula for an ellipsoidal Earth - accurate enough for navigation and surveying at scales below a few thousand kilometres.
Azimuth vs bearing vs quadrant notation
Three conventions exist for expressing a horizontal direction, and they are frequently confused:
- Azimuth (0-360 deg): clockwise from North. Used in most navigation, GIS, and military contexts. Unambiguous and easy to compute with.
- Compass bearing (-180 to +180 deg): the same angle but with East as positive and West as negative. Some software packages and older charts use this signed convention.
- Quadrant bearing (e.g. N45.50E, S12.30W): the angle measured from North or South toward East or West, always between 0 and 90 deg. Preferred in land surveying and legal property descriptions because it makes the general direction obvious at a glance.
This calculator converts freely between all three systems. The back-azimuth is simply the forward azimuth plus 180 degrees, normalized to 0-360. It is the direction you would face to look back along the same line to the origin.
Magnetic declination and true vs magnetic north
A magnetic compass points to magnetic North, which differs from true (geographic) North by an angle called the magnetic declination. This angle varies by location and changes slowly over time. In New York City it is about -12 deg (westward), meaning magnetic North is 12 deg west of true North. In eastern Australia it is around +12 deg eastward. If you enter the declination for your location, the calculator outputs the magnetic azimuth you would set on your compass to follow the calculated true azimuth. Positive declination means magnetic North is east of true North (subtract from true azimuth to get magnetic); negative means it is west (add the magnitude). The World Magnetic Model maintained by NOAA provides up-to-date declination values by location.
Practical applications and limitations
Azimuth calculations are used in:
- Navigation: setting a compass course between two waypoints.
- Surveying: traversing a property boundary using quadrant bearings and distances.
- Astronomy: pointing a telescope - altitude and azimuth together define any point in the sky.
- Solar energy: orienting solar panels for maximum sun exposure requires the sun's azimuth at key times of day.
- Aviation and maritime: flight plans and ship courses are defined by heading (a form of azimuth) and distance.
This calculator uses the spherical-Earth model, which is sufficient for most practical purposes. For sub-centimetre-accurate geodesy, use the Vincenty formula on the WGS84 ellipsoid. Coordinates must be in decimal degrees; to convert from degrees-minutes-seconds (DMS), add minutes/60 and seconds/3600 to the degree value.
Azimuth to compass direction reference
| Azimuth range (deg) | Direction | Abbreviation |
|---|---|---|
| 348.75 - 360 / 0 - 11.25 | North | N |
| 11.25 - 33.75 | North-northeast | NNE |
| 33.75 - 56.25 | Northeast | NE |
| 56.25 - 78.75 | East-northeast | ENE |
| 78.75 - 101.25 | East | E |
| 101.25 - 123.75 | East-southeast | ESE |
| 123.75 - 146.25 | Southeast | SE |
| 146.25 - 168.75 | South-southeast | SSE |
| 168.75 - 191.25 | South | S |
| 191.25 - 213.75 | South-southwest | SSW |
| 213.75 - 236.25 | Southwest | SW |
| 236.25 - 258.75 | West-southwest | WSW |
| 258.75 - 281.25 | West | W |
| 281.25 - 303.75 | West-northwest | WNW |
| 303.75 - 326.25 | Northwest | NW |
| 326.25 - 348.75 | North-northwest | NNW |
Standard 16-point compass rose: azimuth ranges and corresponding abbreviations.
Frequently asked questions
What is the difference between azimuth and bearing?
Azimuth is measured clockwise from North and ranges from 0 to 360 degrees, giving a single unambiguous number for any direction. Bearing can refer to the same thing, but the term is also used for the signed convention (-180 to +180) or for quadrant notation (e.g. N45E). In everyday navigation the two terms are often used interchangeably for the 0-360 convention.
How do I calculate azimuth from two GPS coordinates?
Use the forward-azimuth formula: convert both latitudes and the longitude difference to radians, compute Y = sin(dLon) x cos(lat2) and X = cos(lat1) x sin(lat2) - sin(lat1) x cos(lat2) x cos(dLon), then take atan2(Y, X) in degrees and normalize to 0-360. This calculator does all of that automatically when you enter the two coordinate pairs.
What is a back-azimuth?
The back-azimuth is the direction from the destination back to the origin - exactly the reverse of the forward azimuth. It equals the forward azimuth plus 180 degrees, normalized to the 0-360 range. If the forward azimuth is 050 deg (NE), the back-azimuth is 230 deg (SW).
What is magnetic declination and why does it matter?
Magnetic declination is the angle between true North (the geographic North Pole) and magnetic North (where a compass needle points). It varies by location - roughly -12 deg in New York and +12 deg in eastern Australia - and changes over time as the Earth's magnetic field shifts. When navigating with a magnetic compass, you must apply the declination to convert the true azimuth from a map or GPS to the compass heading you actually walk.
What is quadrant bearing notation?
Quadrant bearing divides the compass into four quadrants (NE, SE, SW, NW) and expresses a direction as an angle from either North or South toward either East or West. For example, N45.5E means 45.5 degrees from North toward East, which equals an azimuth of 45.5 deg. S30W means 30 degrees from South toward West, which equals an azimuth of 210 deg. Land surveyors and legal deeds commonly use this notation.
How accurate is the great-circle distance calculation?
The Haversine formula used here assumes a perfectly spherical Earth with radius 6,371 km. The real Earth is an oblate ellipsoid, so for very long distances (over 1,000 km) the error can reach about 0.3% compared to the Vincenty formula. For most navigation and planning purposes this is negligible - an error of 3 km on a 1,000 km route.
Can I use this calculator for sun azimuth?
This tool calculates the azimuth between two geographic points, not the position of the Sun. Sun azimuth requires the observer's location, date, and time, and uses a separate solar position algorithm (e.g. the NREL SPA). For solar panel orientation or shadow analysis, use a dedicated solar azimuth calculator.