Great Circle Calculator - Distance, Bearing & Midpoint
Enter the latitude and longitude of two points on Earth to calculate the shortest great-circle distance between them, the initial and final bearing, the midpoint coordinates, and an estimated flight time. Choose kilometres, miles, or nautical miles. Results update as you type.
Formula
Worked example
London (51.5074° N, 0.1278° W) to New York (40.7128° N, 74.0060° W): delta-phi = -10.7946°, delta-lambda = -73.8782°. a = sin²(-5.3973°) + cos(51.5074°) cos(40.7128°) sin²(-36.9391°) = 0.0088 + 0.4808 = 0.2267. c = 2 atan2(0.4761, 0.8792) = 0.8729 rad. d = 6371 x 0.8729 = 5,561 km.
What is a great circle?
A great circle is any circle drawn on a sphere whose centre coincides with the centre of the sphere. On Earth, the equator and all lines of longitude are great circles; parallels of latitude (other than the equator) are not. The arc of a great circle between two points is always the shortest possible surface path between them, which is why it is the basis for long-haul flight routes and ocean passages. On a flat Mercator map a great-circle path appears as a curve bowing toward the nearer pole, even though it is the most direct route.
The Haversine formula
The Haversine formula calculates the great-circle distance between two points given their latitudes and longitudes. It was developed in the 19th century to handle the numerical instability that the spherical law of cosines encounters for very short distances. The formula first computes a = sin²(delta-phi/2) + cos(phi1) x cos(phi2) x sin²(delta-lambda/2), where phi is latitude and lambda is longitude. The central angle c = 2 atan2(sqrt(a), sqrt(1-a)), and the distance is d = R x c, where R is Earth's mean radius (6371.0088 km). For most navigation purposes the Haversine result is accurate to within 0.5%; the more complex Vincenty formula on an ellipsoid is needed only for survey-grade precision.
Initial and final bearings
Because the Earth is a sphere, the compass heading from Point A to Point B changes continuously along a great-circle route. The initial bearing is the direction you set off in when leaving Point A; the final bearing is the heading at which you arrive at Point B. On a short domestic flight the two may differ by only a few degrees; on a transoceanic route the difference can exceed 90°. For practical navigation, pilots and mariners either fly/sail a series of rhumb-line (constant-bearing) segments that approximate the great circle, or use autopilot systems that continuously adjust heading. The initial bearing formula is theta = atan2(sin(delta-lambda) x cos(phi2), cos(phi1) x sin(phi2) - sin(phi1) x cos(phi2) x cos(delta-lambda)), normalised to 0-360°. The final bearing is the reverse: find the initial bearing from B to A and add 180°.
Great circle vs. rhumb line
A rhumb line (loxodrome) is a path that crosses every meridian at the same angle, so the compass heading stays constant throughout the journey. It is simple to follow but longer than the great-circle route for most paths. For east-west routes near the equator the two paths are nearly identical, but for routes at higher latitudes the great circle can be significantly shorter. A flight from London to Tokyo, for example, is roughly 1,600 km shorter on the great circle than on the closest rhumb line. Commercial airlines therefore fly great-circle routes, plotting intermediate waypoints to keep the track close to the theoretical arc.
Great-circle distance reference routes
| Route | Distance (km) | Distance (mi) | Initial bearing |
|---|---|---|---|
| London - New York | 5,570 | 3,461 | 288° NW |
| New York - Los Angeles | 3,940 | 2,448 | 272° W |
| London - Sydney | 16,993 | 10,562 | 60° NE |
| Los Angeles - Tokyo | 8,815 | 5,479 | 320° NW |
| New York - London | 5,570 | 3,461 | 51° NE |
| Paris - Beijing | 8,216 | 5,105 | 48° NE |
| Sydney - Johannesburg | 11,052 | 6,868 | 264° W |
| Dubai - Singapore | 5,839 | 3,628 | 96° E |
Approximate great-circle distances for well-known city pairs.
Frequently asked questions
Why is the great-circle route the shortest path on Earth?
Any arc of a great circle is a geodesic on a sphere, meaning it has no curvature relative to the surface itself. Just as a straight line is the shortest path on a flat plane, a great-circle arc is the shortest path on a spherical surface. Any route that deviates from the great circle covers more distance because it adds curvature that is not required by the direct geometry.
Why does the initial bearing differ from the final bearing?
A great-circle path is not a straight line on a Mercator map; it curves toward the pole. As you travel along it your compass heading relative to north changes continuously. On a short route the change is tiny, but on a transoceanic flight it can exceed 90°. The initial bearing is what you set out on at Point A; the final bearing is the heading on arrival at Point B.
What is the difference between a great circle and a rhumb line?
A great circle is the shortest surface path between two points on a sphere, but the compass heading changes along the route. A rhumb line maintains a constant compass heading, making it easier to navigate, but it is a longer path for most routes. Near the equator for east-west travel the two are almost the same; at higher latitudes the great circle can be hundreds or thousands of kilometres shorter.
How accurate is the Haversine formula?
The Haversine formula assumes Earth is a perfect sphere with a radius of about 6,371 km. In reality Earth is an oblate spheroid, slightly flattened at the poles. The spherical assumption introduces an error of up to about 0.5% (roughly 30 km for a 6,000 km route). For survey-grade work, use the Vincenty formula on the WGS 84 ellipsoid, which is accurate to within millimetres. For flight planning and general geography the Haversine result is entirely sufficient.
How do I enter coordinates in decimal degrees?
Decimal degrees express latitude and longitude as a single number with a decimal fraction rather than degrees, minutes, and seconds. Positive latitude is North (negative is South); positive longitude is East (negative is West). To convert from degrees-minutes-seconds, use: decimal = degrees + minutes/60 + seconds/3600. For example, 51° 30' 27" N becomes 51.5075°. Most GPS devices and mapping apps can display coordinates in decimal degrees.