Great circle distance calculator
A great circle follows the shortest route across a spherical surface. Enter the two endpoints below; the result gives distance, departure and arrival bearings, and the route on three map projections.
Great-circle distance between two coordinates
Enter two latitude–longitude pairs to find the shortest spherical surface distance. North and east are positive; south and west are negative.
Shortest spherical surface arc between the two specified endpoints.
The same surface arc expressed in statute miles.
Angles preserved locally; route bends. Latitudes beyond 85.05° are clipped.
Areas preserved; longitude and latitude stretching differ.
Distances from the North Pole preserved; other routes distort.
Working tape
- Start latitude in radians
51.5074 × 0.017453 = 0.898974 - End latitude in radians
48.8566 × 0.017453 = 0.852709 - Longitude difference
2.3522 − (-0.1278) = 2.48 - Longitude difference in radians
2.48 × 0.017453 = 0.043284 - Sine start latitude
sin(0.898974) = 0.782689 - Sine end latitude
sin(0.852709) = 0.753065 - Cosine start latitude
cos(0.898974) = 0.622414 - Cosine end latitude
cos(0.852709) = 0.657946 - Cosine longitude difference
cos(0.043284) = 0.999063 - Product of latitude sines
0.782689 × 0.753065 = 0.589416 - Product of latitude cosines and longitude cosine
0.622414 × 0.657946 × 0.999063 = 0.409131 - Spherical unit-vector dot product
0.589416 + 0.409131 = 0.998546 - Roundoff lower bound
max(-1, 0.998546) = 0.998546 - Roundoff upper bound
min(1, 0.998546) = 0.998546 - Shortest central angle
acos(0.998546) = 0.053925 - Great-circle surface distance
6,371.0084 × 0.053925 = 343.556513 - Convert kilometres to miles
343.556513 ÷ 1.609344 = 213.47612
What a map line measures
The shortest surface route can look curved on a rectangular world map. Longitude wraps at the date line: 179° east and 179° west are only 2° apart on the equator.
This spherical distance uses a mean Earth radius of 6,371.0084 km. A flown journey can be longer because its waypoints do not follow the same arc.
Compare the same endpoints on a polar flat-Earth map →A real southern connection: Sydney–Santiago
Qantas lists a connection between Sydney and Santiago in its international network. Choose the Sydney → Santiago preset to compare the two city-centre coordinates. These are geographic endpoints, not aircraft tracking points or airport survey coordinates.
Compare that shortest sphere arc with the straight segment between the same locations on a north-polar equidistant map. A map projection can enlarge the separation of southern places even when each radial distance from its centre is correct. Flight tracks, departure/arrival routing and winds explain differences between the spherical shortest path and an actual journey.
Qantas international network · Enter the same coordinates in the sphere–polar-map comparison →
A tracked flight: Buenos Aires–Darwin
For QF14 on 5–6 October 2021, the tracking provider reported 14,683 km for the shortest great-circle route and 15,037 km actually flown: 354 km, or about 2.4%, farther. It recorded departure at 15:44 UTC and arrival at 09:10 UTC the following day. The flight reached 74.992844° south at 22:35:51 UTC.
These are the provider’s reported flight measurements, distinct from the city-centre examples above. Its route report describes diversions from the shortest arc for operational limits and weather. A comparison must specify endpoints, a physical map scale and the same journey; a straight line on a projected map is a separate prediction.
Flightradar24’s QF14 tracking report · Compare coordinate distances on the sphere and north-polar plane →