Great circle distance and bearing calculator

The shortest path between two points on a sphere is a great circle, which on a Mercator chart looks like a curve.

From latitude

From longitude

To latitude

To longitude

Ground speed

Great circle distance

5540.02km

≈ Rounded Great circle distance Show in

Show the working

From latitude = 51.47
From longitude = -0.4543
To latitude = 40.6413
To longitude = -73.7781
Ground speed = 450
phi1 = rad(51.47) = 0.898 321
phi2 = rad(40.6413) = 0.709 324
dphi = rad(40.6413 - 51.47) = -0.188 996
dlambda = rad(-73.7781 - -0.4543) = -1.279 74
a = sin(-0.188 996/2)^2 + cos(0.898 321)*cos(0.709 324)*sin(-1.279 74/2)^2 = 0.177 421
c_angle = 2 * atan2(sqrt(0.177 421), sqrt(max(1 - 0.177 421, 0))) = 0.869 567
Great circle distance = 0.869 567 * 6371.0088 = 5540.02
In nautical miles = 5540.02 / 1.852 = 2991.37
In statute miles = 5540.02 / 1.609344 = 3442.41
Initial bearing = mod(deg(atan2(sin(-1.279 74)*cos(0.709 324), cos(0.898 321)*sin(0.709 324) - sin(0.898 321)*cos(0.709 324)*cos(-1.279 74))), 360) = 287.943
flight_hours = 450 > 0 ? 2991.37 / 450 : 0 = 6.647 49
Flight time at that ground speed = 6.647 49 * 3600 = 23 931
Angular separation = deg(0.869 567) = 49.8225

Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.

Scenarios & local history

The formula

haversine: a = sin^2(dphi/2) + cos(phi1)cos(phi2)sin^2(dlambda/2)
distance = 2 R atan2(sqrt(a), sqrt(1-a)) with R the mean Earth radius, 6371.0088 km
a nautical mile is exactly 1852 m, originally one minute of arc

The haversine formula assumes a spherical Earth and is accurate to about 0.3%. Vincenty on the WGS84 ellipsoid is better for surveying.

The initial bearing changes continuously along a great circle. A constant-bearing rhumb line is longer but easier to steer.

This is a planning and study tool. Use certified charts, an approved flight manual and official publications for anything operational.