Great Circle Distance Calculator
Free Great circle distance Converter for geography & distance units. Enter a value to see equivalent measurements across systems.
Great Circle Distance Calculator
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Formula: d = 2R * arcsin(sqrt(sin((lat2-lat1)/2)^2 + cos(lat1)*cos(lat2)*sin((lon2-lon1)/2)^2))
Worked example — Distance: 5,570.25 km | 3,461.05 miles | 3,007.69 nautical miles | Bearing: 51.37 degrees
Formula
d = 2R * arcsin(sqrt(sin((lat2-lat1)/2)^2 + cos(lat1)*cos(lat2)*sin((lon2-lon1)/2)^2))
The Haversine formula calculates the great circle distance by first computing the haversine of the central angle between two points. It uses latitude and longitude in radians, with R being Earth's mean radius of 6,371 km. The formula handles antipodal points correctly and avoids numerical issues at small distances.
Worked Examples
Example 1: New York to London Flight Distance
Problem:Calculate the great circle distance from New York (40.7128 N, 74.006 W) to London (51.5074 N, 0.1278 W).
Solution:Using Haversine formula with R = 6,371 km: dLat = 10.7946 deg, dLon = 73.8782 deg a = sin(5.3973)^2 + cos(40.7128) * cos(51.5074) * sin(36.9391)^2 c = 2 * atan2(sqrt(a), sqrt(1-a)) Distance = 6371 * c
Result:Distance: 5,570.25 km | 3,461.05 miles | 3,007.69 nautical miles | Bearing: 51.37 degrees
Example 2: Sydney to Tokyo
Problem:Find the distance from Sydney (-33.8688, 151.2093) to Tokyo (35.6762, 139.6503).
Solution:dLat = 69.545 deg, dLon = -11.559 deg Apply Haversine formula with Earth radius 6,371 km. Compute intermediate value a, then angular distance c.
Result:Distance: approximately 7,823 km | 4,861 miles | Initial bearing: 338.68 degrees
Frequently Asked Questions
What is the great circle distance?
The great circle distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere rather than through it. On Earth, this represents the shortest flight path between two locations. The concept comes from the fact that the shortest path on a sphere always lies along a great circle, which is a circle whose center coincides with the center of the sphere. Airlines use great circle routes to minimize fuel consumption and flight time.
What is the Haversine formula?
The Haversine formula is a mathematical equation used to calculate the great circle distance between two points on a sphere given their latitude and longitude. It uses the haversine function, which is defined as hav(theta) = sin squared of theta divided by 2. The formula is particularly well-conditioned for numerical computation, avoiding floating point errors that can occur with other spherical trigonometry formulas at small distances. It assumes a perfect sphere, so results may differ slightly from real Earth distances due to the oblate shape of our planet.
How accurate is the Haversine formula for Earth distances?
The Haversine formula typically provides accuracy within about 0.3% of the true distance because it models Earth as a perfect sphere with radius 6,371 km. In reality, Earth is an oblate spheroid with an equatorial radius about 21 km larger than the polar radius. For higher accuracy, the Vincenty formula uses an ellipsoidal model and provides accuracy to within 0.5 mm. For most practical purposes including aviation and navigation, the Haversine formula provides more than sufficient accuracy.
What is the initial bearing in navigation?
The initial bearing, also called forward azimuth, is the compass direction you would need to travel from the starting point to reach the destination along the great circle path. Unlike a rhumb line which maintains a constant bearing, the bearing along a great circle route continuously changes as you travel. The initial bearing is measured in degrees clockwise from true north, where 0 degrees is north, 90 degrees is east, 180 degrees is south, and 270 degrees is west.
Why do flight paths shown on airline maps look curved instead of straight?
The curve you see on an in-flight map or airline route diagram is not a detour, it is the great circle route rendered onto a flat map projection, which distorts straight spherical paths into arcs. Most world maps use a Mercator-style projection that preserves angles and shapes reasonably well near the equator but stretches distances and directions dramatically near the poles, so a genuinely shortest path between two mid-to-high-latitude cities (say, New York to Tokyo) appears to bow northward toward the Arctic on the map even though it is the most direct route by actual flown distance. This is also why a globe, not a flat map, is the only projection where a great circle route always appears as a straight line.
What is the difference between the Haversine formula and Vincenty's formulae?
Haversine treats the Earth as a perfect sphere of a single mean radius, which makes it short, numerically stable and fast. Vincenty's formulae model the Earth as an oblate ellipsoid such as WGS-84, iterating to solve for the geodesic between two points, and can agree with that reference shape to millimetre level. The price is complexity and slow or failed convergence for nearly antipodal points, which Karney's later algorithm addresses. For route planning and distance tables Haversine is normally adequate; for surveying and geodetic control, use an ellipsoidal method.
How does a rhumb line differ from a great circle route?
A rhumb line, or loxodrome, crosses every meridian at the same angle, so you steer one constant compass bearing from start to finish. It plots as a straight line on a Mercator chart, which is exactly why that projection was devised. A great circle is the shortest path on the sphere but requires the heading to change continuously. On short legs the difference is negligible, but across an ocean at high latitude a rhumb line can be noticeably longer. Traditional practice splits a great circle into a series of rhumb-line legs between waypoints.
Why does the initial bearing change as you travel along a great circle?
Because meridians are not parallel; they converge toward the poles. A great circle keeps a constant plane through the Earth's centre, but the angle at which that plane cuts successive meridians changes as latitude changes, so the true course drifts continuously. The two exceptions are the equator and any meridian, where the great circle and the constant-bearing track coincide. This is why the calculator reports an initial bearing rather than a single course, and why the final bearing at the destination differs from the one you set out on.
How do you convert a great circle angular distance into nautical miles?
The central angle between two points, expressed in degrees and multiplied by 60, gives the distance in nautical miles almost directly, because the nautical mile was originally defined as one arcminute of a great circle on the Earth's surface. The modern definition fixes it at exactly 1,852 metres, a rounded average since the Earth is not a perfect sphere. So a central angle of 30 degrees is 1,800 arcminutes, or about 1,800 nautical miles, roughly 3,334 kilometres. This relationship is why the latitude scale on a nautical chart doubles as a distance scale.
Which Earth radius should be used, and how much does the choice matter?
Haversine calculations normally use the mean radius of about 6,371 kilometres. The equatorial radius is 6,378.137 kilometres and the polar radius about 6,356.752, a spread of roughly 0.34 percent caused by a flattening of about 1/298.257. Using the mean radius therefore leaves a distance error of a few tenths of a percent depending on the latitudes involved, which is a few kilometres on a transatlantic leg. If your route stays within one latitude band, a radius appropriate to that band reduces the error; if you need better than that, switch to an ellipsoidal geodesic.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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