Great Circle Calculator
Free Great circle Calculator for 3d geometry. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
d = R * 2 * atan2(sqrt(a), sqrt(1-a))
Where a = sin^2(dLat/2) + cos(lat1) * cos(lat2) * sin^2(dLon/2), R is the sphere radius (6,371 km for Earth), and d is the great circle distance. This is the Haversine formula, which computes the central angle between two points and multiplies by the radius to get arc length.
Worked Examples
Example 1: New York to London Flight Distance
Problem:Calculate the great circle distance from New York (40.7128 N, 74.0060 W) to London (51.5074 N, 0.1278 W).
Solution:Using the Haversine formula with Earth radius = 6,371 km:\ndPhi = (51.5074 - 40.7128) = 10.7946 degrees\ndLambda = (-0.1278 - (-74.006)) = 73.8782 degrees\na = sin^2(5.3973) + cos(40.7128) x cos(51.5074) x sin^2(36.9391)\nc = 2 x atan2(sqrt(a), sqrt(1-a)) = 0.8676 radians\nDistance = 6371 x 0.8676 = 5,527.2 km\nBearing from NYC = approximately 51.2 degrees (northeast)
Result:Distance: 5,527.2 km (3,434.3 mi) | Initial bearing: 51.2 deg | Flight time: ~6.1 hrs
Example 2: Sydney to Santiago Transpacific Route
Problem:Find the great circle distance from Sydney (-33.8688, 151.2093) to Santiago (-33.4489, -70.6693).
Solution:Using Haversine with R = 6,371 km:\ndPhi = (-33.4489 - (-33.8688)) = 0.4199 degrees\ndLambda = (-70.6693 - 151.2093) = -221.8786 degrees (adjusted to 138.1214)\na = sin^2(0.2100) + cos(-33.8688) x cos(-33.4489) x sin^2(69.0607)\nc = 2 x atan2(sqrt(a), sqrt(1-a)) = 1.7505 radians\nDistance = 6371 x 1.7505 = 11,151.4 km
Result:Distance: 11,151.4 km (6,927.8 mi) | Flight time: ~12.4 hrs jet
Frequently Asked Questions
What is a great circle and why is it important?
A great circle is the largest possible circle that can be drawn on the surface of a sphere, formed by the intersection of the sphere with a plane that passes through the center of the sphere. The equator is a great circle on Earth, as are all lines of longitude (meridians). The great circle path between any two points on a sphere represents the shortest distance between those points along the surface, which is why airplanes and ships follow great circle routes for long-distance travel. This shortest path is called a geodesic. Unlike straight lines on a flat map, great circle routes often appear curved on common map projections like the Mercator projection. Understanding great circles is fundamental to navigation, aviation, telecommunications satellite coverage, and spherical geometry.
How does the Haversine formula calculate great circle distance?
The Haversine formula calculates the shortest distance between two points on a sphere using their latitude and longitude coordinates. The formula first computes the central angle between the two points using the haversine function, which is defined as hav(theta) = sin squared (theta/2). The specific formula is: a = sin squared((lat2-lat1)/2) + cos(lat1) times cos(lat2) times sin squared((lon2-lon1)/2), then c = 2 times atan2(sqrt(a), sqrt(1-a)), and finally d = R times c, where R is the sphere radius. The haversine formula is preferred over the law of cosines for small distances because it remains numerically stable even when the two points are very close together, avoiding the floating-point arithmetic problems that plague the cosine formula at short distances.
What is the initial bearing and why does it change along a great circle?
The initial bearing, also called the forward azimuth, is the compass direction you need to travel from the starting point to follow the great circle route. Unlike on a flat surface where direction remains constant along a straight line, the bearing continuously changes along a great circle path because of the curvature of the sphere. For example, a flight from London to New York starts heading roughly southwest but gradually shifts to a more westerly and then more westerly-to-southwesterly direction. The bearing formula uses arctangent of the ratio of east-west and north-south components, calculated from the coordinates of both points. This changing bearing is why navigators historically had to constantly adjust their heading during long ocean voyages. Modern autopilot systems continuously recalculate the bearing to maintain the great circle path.
How accurate is great circle distance for real Earth navigation?
Great circle calculations assuming a perfect sphere are accurate to within about 0.5 percent for most practical navigation purposes. The actual Earth is an oblate spheroid, slightly flattened at the poles with an equatorial radius of 6,378.137 km and a polar radius of 6,356.752 km. For higher precision, the Vincenty formula or Karney method use the WGS-84 ellipsoid model and achieve accuracy within 0.5 millimeters for any distance. For aviation, the spherical great circle calculation is more than adequate since winds, routing around restricted airspace, and altitude variations introduce far larger uncertainties than the spherical approximation error. For geodetic surveying and precision mapping, ellipsoidal calculations are essential. Great Circle Calculator uses the spherical model with a default radius of 6,371 km, which is the mean radius of Earth.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy