Distance Between Coordinates

How to calculate the distance between two coordinates

1. Type the Latitude and Longitude of Point A and Point B. Decimal degrees (48.8566, -74.006) and degrees, minutes, seconds (48°51‘24”N, 48 51 24 N, 48°51.4’N) both work. South latitudes and west longitudes are negative, or you can add the letters N, S, E and W instead of a sign.
2. If you copied a pair such as “48.8566, 2.3522” from a map, paste it into a latitude box: this tool splits it and fills both boxes.
3. Choose the Method: Haversine (sphere) or Vincenty (WGS84 ellipsoid), and the Unit used for the main result.
4. The result updates as you type (Calculate does the same). You get the Distance in kilometres, miles and nautical miles, the Initial bearing, the Final bearing and the Midpoint of the route, in decimal degrees and in DMS.
5. The line under the main result always compares both methods, so you can see how much the shape of the Earth matters for your two points. Swap A ⇄ B reverses the route, Copy copies the distance, bearing and midpoint, and Step by step shows the working.

The grid below the results draws the great-circle route on a simple latitude/longitude grid (a plate carrée projection). On that flat grid the shortest route usually looks curved: a flight from Paris to New York heads west-north-west, bends towards Newfoundland and arrives heading south-west.

Haversine: the distance on a sphere

The haversine formula gives the great-circle distance, the shortest path along the surface of a sphere. With latitudes φ and longitudes λ in radians:

• a = sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)
• c = 2 · atan2(√a, √(1 − a))
• d = R · c

This tool uses R = 6,371.0088 km, the IUGG mean Earth radius. Because the real Earth is slightly flattened, a spherical result can be off by up to about 0.5 %, which is fine for most travel, logistics and trivia questions. Paris (48.8566, 2.3522) to New York (40.7128, -74.006) comes out at about 5,837 km with haversine.

Vincenty: the distance on the WGS84 ellipsoid

GPS coordinates are defined on the WGS84 ellipsoid (semi-major axis a = 6,378,137 m, flattening f = 1/298.257223563). Vincenty’s inverse formula solves the shortest path on that ellipsoid by iteration and is accurate to within a millimetre or so. For Paris to New York it gives about 5,853 km, some 16 km more than the sphere. The difference between the methods depends on the direction: routes along a meridian and routes along the equator are affected differently.

Vincenty’s iteration can fail to converge for nearly antipodal points (two places almost exactly opposite on the globe). In that rare case this tool shows a note and falls back to the haversine result.

Bearing and midpoint

The initial bearing (or forward azimuth) is the direction you start in, measured clockwise from true north: 0° is north, 90° east, 180° south, 270° west. On a great circle the direction changes along the way, which is why a final bearing is also shown: it is the direction of travel when you arrive at B. Both are true bearings; to steer with a magnetic compass you still need to apply the local magnetic declination.

The midpoint is the point halfway along the route, not the average of the coordinates. With Vincenty it is found by travelling half the distance along the ellipsoidal geodesic (Vincenty’s direct formula).