Azimuth Angle Calculator
Our free geography & distance converter handles azimuth angle conversions. See tables, ratios, and examples for quick reference.
Azimuth Angle Calculator
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.
How do you turn a true azimuth into a compass heading?
Add or subtract the magnetic declination for your location, the angle between true north and magnetic north. Where declination is east, magnetic azimuth equals true azimuth minus declination; where it is west, you add it. Declination varies from place to place and drifts year on year as the geomagnetic field changes, so navigators take it from a current chart or a field model rather than memory. The azimuths computed here are true, referenced to the geographic pole, not to a compass needle.
Why does the bearing formula use atan2 instead of a plain arctangent?
An ordinary arctangent of the ratio y over x collapses two pieces of information into one and returns only values between -90 and +90 degrees, so a heading of 200 degrees comes back indistinguishable from 20. The atan2 function takes the numerator and denominator separately, keeps both signs, and resolves the correct quadrant across the whole circle. It also handles a zero denominator, which occurs when the destination lies due north or due south and a plain division would fail.
Why is the back azimuth not simply the forward azimuth plus 180 degrees?
That rule holds on a flat plane, and on a sphere only in the special cases of travel along a meridian or along the equator. Elsewhere the meridians converge toward the poles, so a great circle crosses successive meridians at steadily changing angles. Leave New York on a heading near 51 degrees for London and you arrive on a course much further round the compass, which is why the return heading must be computed from the reversed pair of points rather than assumed.
How does an azimuth become a compass point like NNE?
The 16-point rose divides the full circle into equal sectors of 360 / 16 = 22.5 degrees, each centred on its named direction. Dividing the azimuth by 22.5 and rounding to the nearest whole number picks the sector, with the count wrapping back to north past 15. So 51.2 degrees divides to 2.28, rounds to 2, and lands on NE. A 32-point rose halves the sectors to 11.25 degrees and adds names such as north-east by north.
How are fractions of a degree written in surveying and artillery?
Surveyors use degrees, minutes and seconds, where a degree holds 60 arcminutes and each minute holds 60 arcseconds. An azimuth of 51.21 degrees is 51 degrees plus 0.21 x 60 = 12.6 arcminutes, so 51 degrees 12 arcminutes 36 arcseconds. Gunnery instead uses the mil, with 6,400 mils to the full circle, so one mil is 0.05625 degrees and the same azimuth is about 910 mils. The mil is popular because at one kilometre it subtends roughly one metre.
Why is the distance also reported in nautical miles?
The nautical mile was originally defined as one arcminute of latitude along a meridian, which ties distance directly to angle: one degree of latitude spans about 60 nautical miles anywhere on the globe. It is now fixed at exactly 1,852 metres, so kilometres convert by dividing by 1.852, and a 5,570 km great circle is about 3,008 nautical miles. Because speed in knots means nautical miles per hour, marine and air navigation keep the unit for direct time-and-distance work.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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