Azimuth Angle Calculator
Our free geography & distance converter handles azimuth angle conversions. See tables, ratios, and examples for quick reference.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Azimuth Angle Calculator
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Formula: Azimuth = atan2(sin(dLon)*cos(lat2), cos(lat1)*sin(lat2) - sin(lat1)*cos(lat2)*cos(dLon))
Worked example — Forward azimuth = ~51.21 degrees (northeast), Distance = ~5,570 km
Formula
Azimuth = atan2(sin(dLon)*cos(lat2), cos(lat1)*sin(lat2) - sin(lat1)*cos(lat2)*cos(dLon))
The azimuth formula uses spherical trigonometry to compute the initial bearing of the great circle path between two points. The atan2 function correctly handles all quadrants, producing a result in radians that is then converted to degrees and normalized to the 0-360 range. The back azimuth is computed by reversing the point order.
Worked Examples
Example 1: New York to London
Problem:Calculate the azimuth from New York (40.7128N, 74.006W) to London (51.5074N, 0.1278W).
Solution:Convert to radians and apply the formula: x = sin(73.8782 deg) * cos(51.5074 deg) = 0.5987 y = cos(40.7128) * sin(51.5074) - sin(40.7128) * cos(51.5074) * cos(73.8782) y = 0.3377 Azimuth = atan2(0.5987, 0.3377) = 51.21 degrees (NE direction)
Result:Forward azimuth = ~51.21 degrees (northeast), Distance = ~5,570 km
Example 2: Surveying Between Points
Problem:Find the azimuth from point A (34.0522N, 118.2437W) to point B (36.1699N, 115.1398W) (Los Angeles to Las Vegas).
Solution:delta_lon = -115.1398 - (-118.2437) = 3.1039 deg x = sin(3.1039 deg) * cos(36.1699 deg) = 0.04376 y = cos(34.0522) * sin(36.1699) - sin(34.0522) * cos(36.1699) * cos(3.1039) y = 0.03726 Azimuth = atan2(0.04376, 0.03726) = 49.57 degrees
Result:Forward azimuth = ~49.57 degrees (NE), Distance = ~368 km
Frequently Asked Questions
What is an azimuth angle?
An azimuth angle is the horizontal angle measured clockwise from true north to the direction of an object or destination. It ranges from 0 to 360 degrees, where 0/360 is north, 90 is east, 180 is south, and 270 is west. Azimuth is fundamental in navigation, surveying, astronomy, and military applications. Unlike a simple compass bearing, azimuth is always measured from north and always clockwise, providing an unambiguous directional reference.
How is the forward azimuth calculated between two coordinates?
The forward azimuth between two geographic coordinates is calculated using the initial bearing formula from spherical trigonometry. The formula uses the arctangent of two values: the numerator is sin(delta_longitude) times cos(latitude2), and the denominator is cos(latitude1) times sin(latitude2) minus sin(latitude1) times cos(latitude2) times cos(delta_longitude). The result is normalized to a 0-360 degree range. This gives the initial direction of the great circle path connecting the two points.
What is the difference between azimuth and bearing?
Azimuth is always measured clockwise from north as a single number between 0 and 360 degrees, such as 225 degrees. Bearing can be expressed in two formats: as a three-digit azimuth (225 degrees) or as a quadrant bearing using cardinal directions like S45W, meaning 45 degrees west of south. Quadrant bearings are common in land surveying and never exceed 90 degrees. True azimuth references geographic north, magnetic azimuth references magnetic north, and grid azimuth references the map grid north.
Why does the azimuth change along a great circle route?
On a sphere, a great circle route is the shortest path between two points, but it does not maintain a constant azimuth because meridians converge toward the poles. The initial azimuth (forward azimuth) gives the heading at the departure point, while the back azimuth gives the arrival heading. The azimuth continuously changes along the route. A rhumb line, by contrast, maintains a constant azimuth but is longer than the great circle distance. This distinction is critical for long-distance air and sea navigation.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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