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 for accuracy by Manoj Kumar, Mathematics Educator
Great Circle Calculator
Calculator
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Formula: d = R * 2 * atan2(sqrt(a), sqrt(1-a))
Worked example — Distance: 5,527.2 km (3,434.3 mi) | Initial bearing: 51.2 deg | Flight time: ~6.1 hrs
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: dPhi = (51.5074 - 40.7128) = 10.7946 degrees dLambda = (-0.1278 - (-74.006)) = 73.8782 degrees a = sin^2(5.3973) + cos(40.7128) x cos(51.5074) x sin^2(36.9391) c = 2 x atan2(sqrt(a), sqrt(1-a)) = 0.8676 radians Distance = 6371 x 0.8676 = 5,527.2 km Bearing 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: dPhi = (-33.4489 - (-33.8688)) = 0.4199 degrees dLambda = (-70.6693 - 151.2093) = -221.8786 degrees (adjusted to 138.1214) a = sin^2(0.2100) + cos(-33.8688) x cos(-33.4489) x sin^2(69.0607) c = 2 x atan2(sqrt(a), sqrt(1-a)) = 1.7505 radians Distance = 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.
Why do airplanes fly curved routes instead of straight lines?
Airplanes appear to fly curved routes on flat maps, but they are actually flying the straightest possible path on the curved surface of the Earth, which is a great circle route. A Mercator projection map distorts distances at higher latitudes, making great circle routes look curved when they are actually the most direct paths. For example, a flight from New York to Tokyo does not follow a seemingly straight west-to-east path across the Pacific. Instead, it flies northwest over Canada and Alaska, which is significantly shorter because the great circle route takes advantage of the fact that the Earth is a sphere. A flight from London to Los Angeles passes over southern Greenland and northern Canada rather than going straight west. These great circle routes can save hundreds of miles, translating to thousands of dollars in fuel savings and hours of flight time.
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.
What is the difference between a great circle and a rhumb line?
A great circle is the shortest path between two points on a sphere, while a rhumb line, also called a loxodrome, is a path of constant bearing that crosses all meridians at the same angle. On a Mercator projection map, a rhumb line appears as a straight line, while a great circle appears curved. However, on the actual sphere, the rhumb line is longer than the great circle except when traveling along the equator or along a meridian, where both paths are identical. Historically, sailors preferred rhumb lines because maintaining a constant compass heading was much simpler than continuously adjusting course to follow a great circle. Modern GPS navigation makes following great circle routes trivial, so the fuel and time savings of the shorter path can be realized. For short distances, the difference between the two paths is negligible.
How do you find the midpoint of a great circle route?
The midpoint of a great circle route is calculated using vector mathematics on the sphere. The formula converts both endpoints from latitude and longitude to three-dimensional Cartesian coordinates, averages the vectors, and converts back to latitude and longitude. Specifically, the midpoint latitude is atan2(sin(lat1) + sin(lat2), sqrt((cos(lat1) + Bx)^2 + By^2)), and the midpoint longitude is lon1 + atan2(By, cos(lat1) + Bx), where Bx = cos(lat2) times cos(dLon) and By = cos(lat2) times sin(dLon). The midpoint is the point at which exactly half the great circle distance has been covered. This is useful for determining diversion airports for long-haul flights, finding the point where a cable or pipeline crosses a particular latitude, and for waypoint navigation. The midpoint on a great circle is not the same as the geographic midpoint on a map projection.
Can great circle calculations be used for spheres other than Earth?
Absolutely. Great circle calculations apply to any sphere, not just Earth. By changing the sphere radius parameter in the calculator, you can compute great circle distances on the Moon (radius 1,737.4 km), Mars (radius 3,389.5 km), Jupiter (radius 69,911 km), or any other spherical body. Planetary scientists use great circle calculations to plan rover paths on Mars, determine communication satellite coverage areas around other planets, and measure distances between surface features on moons. The formulas work identically regardless of sphere size because they depend only on the central angle and radius. Astronomers also use great circle concepts on the celestial sphere, where angular distances between stars and other celestial objects are measured as great circle arcs. In computer graphics, great circle interpolation is used for smooth camera rotation.
What is the antipodal point and how does it relate to great circles?
The antipodal point of any location on a sphere is the point diametrically opposite to it, exactly on the other side of the sphere. Every great circle passing through a given point also passes through its antipodal point, and the great circle distance to the antipodal point is exactly half the circumference of the sphere, which equals pi times the radius. For Earth, this maximum possible great circle distance is approximately 20,015 km. The antipodal point of any location has a latitude equal to the negative of the original latitude and a longitude differing by 180 degrees. Most land areas on Earth have their antipodal points in the ocean, so it is relatively rare for two major landmasses to be antipodal. The most notable exception is parts of Spain and New Zealand, which are approximately antipodal. All great circles through a point form a pencil of circles sharing the point and its antipode.
How do nautical miles relate to great circle navigation?
Nautical miles have a direct mathematical relationship to great circle navigation that makes them extremely convenient for maritime and aviation use. One nautical mile is defined as exactly one minute (one-sixtieth of a degree) of arc along a great circle on Earth. This means that the great circle distance between two points can be directly read from a nautical chart by measuring the angular difference in latitude along a meridian. One degree of great circle arc equals 60 nautical miles. The total Earth circumference along any great circle is approximately 21,600 nautical miles (360 degrees times 60 minutes). One nautical mile equals 1.852 kilometers or 1.15078 statute miles. Speed measured in nautical miles per hour is called knots, making it easy to convert between speed, distance, and angular displacement. This is why the nautical mile remains the standard unit of distance in aviation and maritime navigation worldwide.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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