How to use the destination point from bearing calculator
- Enter the start latitude and longitude in decimal degrees, with south and west as negative numbers.
- Enter the bearing, measured clockwise from north (0 north, 90 east, 180 south, 270 west), and the distance in kilometers, miles or nautical miles.
- Read the destination, its degrees minutes and seconds, the bearing on arrival and the check distance back.
- Copy the table or download the CSV. Change the numbers to see how the destination moves.
Formula
δ = distance / R, with R = 6371.0088 km
φ2 = asin(sin φ1 cos δ + cos φ1 sin δ cos θ)
λ2 = λ1 + atan2(sin θ sin δ cos φ1, cos δ - sin φ1 sin φ2)
bearing on arrival = bearing from the destination back to the start + 180°
φ1 and λ1 are the start latitude and longitude, θ is the bearing and δ is the angular distance. The path followed is a great circle, the shortest route over a sphere, so the bearing changes along the way.
Due east from New York
Starting at 40.7128° N, 74.0060° W and heading due east (bearing 90°) for 500 km, the destination is 40.56121° N, 68.08270° W. The latitude falls slightly even though the heading was due east, because a great circle curves toward the equator when it starts out heading east in the northern hemisphere.
The bearing on arrival is 93.86°, a little south of east, for the same reason. The haversine distance from the destination back to the start is 500 km, which confirms the calculation.
Great circles and constant bearings
A constant compass bearing does not follow a great circle except due north or south and along the equator. Over short distances the two are close, but over thousands of kilometers they differ a lot. This page follows the great circle that leaves the start on the bearing you give.
Because the Earth is slightly flattened, the sphere model can differ from an ellipsoidal one by up to about 0.5 percent of the distance. That is fine for planning and teaching, and it is not enough for surveying or navigation.
Crossing the date line and poles
The destination longitude is brought back into the range from -180 to 180, so a trip east from 178° E shows a negative longitude after it crosses the date line. The distance is limited to half the way around the Earth, because beyond that the path would pass its opposite point and the round-trip check would no longer apply.
Find the distance and bearing between two known points with the Great Circle Distance Calculator, convert bearings with the Bearing and Azimuth Converter, and change coordinate formats with the Coordinate Format Converter.
Assumptions and limits
- The Earth is a sphere with a radius of 6,371.0088 km, the mean radius.
- Latitude runs from -90 to 90 and longitude from -180 to 180. The bearing is from 0 to 360 degrees from true north.
- The distance can be up to about 20,015 km (half the circumference). Miles are statute miles of 1.609344 km, and a nautical mile is 1.852 km.
- This is for planning and study, not navigation, aviation or surveying.
- Coordinates stay in this browser.
Frequently asked questions
Why does heading due east change my latitude?
A great circle is not a line of constant latitude. Starting out due east, it bends toward the equator, so the latitude drops a little over a long distance in the northern hemisphere.
What is the bearing on arrival?
It is the direction you would be heading when you reach the destination, measured clockwise from north. On a great circle it differs from the starting bearing, and the page shows both.
How accurate is it?
It is exact for a perfect sphere. Against the real, slightly flattened Earth the error can reach about half a percent of the distance, which is about 2.5 km in 500 km.
Why is the round trip check useful?
It uses a different formula, the haversine, to measure the distance from the destination back to the start. If it matches the distance you entered, the destination is consistent with the other formulas.