Bearing Calculator
Our free geography & distance converter handles bearing conversions. See tables, ratios, and examples for quick reference.
Bearing Calculator
Calculator
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Formula: Bearing = atan2(sin(dLon)*cos(lat2), cos(lat1)*sin(lat2) - sin(lat1)*cos(lat2)*cos(dLon))
Worked example — True bearing = ~125.56 deg (SE direction)
Formula
Bearing = atan2(sin(dLon)*cos(lat2), cos(lat1)*sin(lat2) - sin(lat1)*cos(lat2)*cos(dLon))
The bearing formula uses spherical trigonometry to compute the initial direction of the great circle path. The atan2 function handles all quadrants correctly, producing a result normalized to 0-360 degrees. Magnetic bearing is derived by subtracting the local magnetic declination from the true bearing.
Worked Examples
Example 1: Paris to Rome
Problem:Calculate the bearing from Paris (48.8566N, 2.3522E) to Rome (41.9028N, 12.4964E).
Solution:delta_lon = 12.4964 - 2.3522 = 10.1442 deg x = sin(10.1442) * cos(41.9028) = 0.1309 y = cos(48.8566) * sin(41.9028) - sin(48.8566) * cos(41.9028) * cos(10.1442) y = -0.0936 Bearing = atan2(0.1309, -0.0936) = 125.56 deg (SE)
Result:True bearing = ~125.56 deg (SE direction)
Example 2: Magnetic Bearing Correction
Problem:A bearing of 90 degrees true with magnetic declination of -5 degrees (west). What is the magnetic bearing?
Solution:Magnetic bearing = True bearing - Declination Magnetic bearing = 90 - (-5) = 95 degrees When declination is west (negative), magnetic north is west of true north.
Result:Magnetic bearing = 95 degrees
Frequently Asked Questions
What is a bearing in navigation?
A bearing is the direction from one point to another, measured as a clockwise angle from north. True bearing is measured from geographic (true) north, while magnetic bearing is measured from magnetic north. Bearings range from 0 to 360 degrees and are typically expressed as three-digit numbers, such as 045 for northeast or 270 for due west. Bearings are essential in marine navigation, aviation, hiking, surveying, and military operations for determining travel direction.
What is the difference between true bearing and magnetic bearing?
True bearing is measured from geographic north, the fixed point where the Earth's rotation axis meets the surface. Magnetic bearing is measured from magnetic north, which varies by location and changes over time due to shifts in Earth's magnetic field. The difference between them is called magnetic declination (or variation). To convert: magnetic bearing equals true bearing minus declination. If declination is east, subtract it; if west, add it. Declination can range from near zero to over 20 degrees depending on location.
How do I convert between azimuth bearing and quadrant bearing?
Quadrant bearing divides the compass into four quadrants and expresses direction as an angle from north or south toward east or west. For example, azimuth 045 becomes N45E, azimuth 135 becomes S45E, azimuth 225 becomes S45W, and azimuth 315 becomes N45W. To convert from quadrant to azimuth: NxE stays the same, SxE becomes 180 minus x, SxW becomes 180 plus x, and NxW becomes 360 minus x. Quadrant bearings are common in land surveying and legal property descriptions.
What is the final bearing and why does it differ from the initial bearing?
The final bearing is the direction of travel when arriving at the destination, while the initial bearing is the direction when departing. On a great circle route across a sphere, the bearing continuously changes because meridians converge at the poles. For short distances the difference is negligible, but for intercontinental travel it can be significant. The final bearing is calculated by computing the back azimuth from destination to origin and then adding 180 degrees to reverse it.
What is the difference between a great circle route and a rhumb line?
A great circle is the shortest path between two points on a sphere, but its heading changes continuously because the meridians it crosses converge toward the poles. A rhumb line, or loxodrome, crosses every meridian at the same angle, so you steer one constant bearing the whole way and it plots as a straight line on a Mercator chart. The trade-off is distance: the rhumb line is always longer, and the penalty grows with route length and latitude.
How is the distance between the two coordinates calculated?
This tool uses the haversine formula on a spherical Earth of mean radius 6,371 km. It computes a = sin squared of half the latitude difference plus cos(lat1) x cos(lat2) x sin squared of half the longitude difference, then the central angle c = 2 x atan2 of the square root of a and the square root of one minus a, and finally distance = R x c. Because the Earth is a slightly flattened ellipsoid, spherical results differ from ellipsoidal ones by a few tenths of a percent.
What is a back bearing and how do I work one out?
A back bearing is the direction from the destination toward the origin. On a flat map or over short distances you simply add 180 degrees if the forward bearing is under 180, or subtract 180 if it is over, keeping the result within 0 to 360. Over long great circle routes that shortcut fails, because meridian convergence means the arrival heading is not the departure heading reversed. That is why Bearing Calculator reports an initial and a final bearing separately.
How should I enter coordinates for locations south of the equator or west of Greenwich?
Use signed decimal degrees: latitude is positive north and negative south, longitude is positive east and negative west. Sydney is about -33.87 latitude and 151.21 longitude, while New York is about 40.71 and -74.01. Dropping a minus sign is the single most common input error, and it can swing the computed bearing by a hundred degrees or more. If your source gives degrees, minutes and seconds, convert first by adding minutes over 60 and seconds over 3600.
Why does magnetic declination change over time?
Magnetic declination is the angle between true north and magnetic north at a specific place. It exists because the geomagnetic field is generated by convecting molten iron in the Earth's outer core, and that flow drifts, so the magnetic poles wander and the field slowly changes. Charts therefore print a declination value together with the year it was measured and an annual rate of change. For accurate work, look up a current value for your location rather than reusing an old chart figure.
When does grid north matter alongside true and magnetic north?
Topographic maps drawn on a projection such as UTM or a national grid have a third reference: grid north, the direction of the vertical grid lines. Grid north differs from true north by the grid convergence, an angle that grows the farther you are from the projection's central meridian and with increasing latitude. Surveyors and military users therefore apply a grid magnetic angle to move between a compass reading and a bearing plotted on the sheet. Bearing Calculator works in true and magnetic bearings only.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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