How to use the great circle distance calculator
- Enter the latitude and longitude of the start and the end point in decimal degrees. North and east are positive, south and west are negative.
- Choose a distance unit: kilometers, miles or nautical miles. All three are shown.
- Read the distance, the initial and final bearing in degrees from north, and the midpoint.
- To use degrees, minutes and seconds, convert them first with the Coordinate Format Converter.
Formula
a = sin^2(dLat / 2) + cos(lat1) x cos(lat2) x sin^2(dLon / 2)
angle = 2 x atan2(sqrt(a), sqrt(1 - a))
distance = angle x 6371.0088 km
initial bearing = atan2(sin(dLon) x cos(lat2), cos(lat1) x sin(lat2) - sin(lat1) x cos(lat2) x cos(dLon))
This is the haversine formula on a sphere with the mean Earth radius. The bearing is measured clockwise from north and runs from 0 to 360 degrees.
New York to London
From 40.7128, -74.0060 to 51.5074, -0.1278 the central angle is 50.0942 degrees, which is 5,570.23 km, 3,461.18 miles or 3,007.68 nautical miles. The initial bearing is 51.21 degrees (northeast), and the final bearing on arrival is 108.33 degrees (east southeast), because a great circle bends relative to the compass lines.
The midpoint is at 52.36844, -41.29031. Sydney to Tokyo (-33.8688, 151.2093 to 35.6762, 139.6503) is 7,825.83 km with an initial bearing of 350.05 degrees. The same distance comes out of a second method that uses vectors from the center of the Earth, with the angle from the cross and dot products.
Why the shortest path looks curved on a map
A great circle is the path you get by cutting a sphere with a plane through its center. It is the shortest route between two points on the surface. On a flat map, most straight lines are not great circles, so routes that look curved on the map are often the shortest, which is why flights between far cities arc toward the poles.
Because a great circle crosses the meridians at changing angles, the heading changes along the way. That is why the initial and final bearings differ. The Flight Arrival Time Calculator and the Road Trip Time Calculator turn distance and speed into trip times.
How accurate is it
The Earth is slightly flattened, not a perfect sphere, so a spherical formula can be off by up to about half a percent compared with a surveyed distance. That is fine for planning, teaching and comparing cities and not suitable for navigation or surveying. Road distance is longer than the straight great circle, sometimes by a lot.
If your points are in another format such as 40 26 46 N, use the Coordinate Format Converter. To compare local times in two places, use the Time Zone Converter.
Assumptions and limits
- The Earth is a sphere with a mean radius of 6,371.0088 km.
- Latitudes run from -90 to 90 and longitudes from -180 to 180 degrees.
- For identical or exactly opposite points the bearing has no single value and the page says so.
- Altitude, terrain, roads, flight paths and wind are not considered.
Frequently asked questions
What is a great circle?
It is a circle on a sphere whose center is the center of the sphere. The shorter arc of a great circle between two points is the shortest path between them over the surface.
Why are the initial and final bearings different?
A great circle does not keep a constant compass heading. It crosses each meridian at a different angle, so the direction you face at the start differs from the direction at the end.
How accurate is the distance?
It is within about half a percent of a distance measured on the real, slightly flattened Earth. For a short trip across town the difference is a few meters, and for long distances it can reach tens of kilometers.
Is a nautical mile different from a mile?
Yes. A nautical mile is 1,852 meters, about one minute of latitude, and is used in air and sea travel. A statute mile is 1,609.344 meters.