Our great circle calculator will show you why you crossed the tip of Greenland while traveling from Los Angeles to London, even though it looks well out of the way. Here you will understand how maps bend reality and acquire a better grasp of aerial routes.
Take off with CalcuGo and discover the quirks of geodesics; keep reading to learn:
- What is the great circle?
- How many great circles are out there?
- What is the great circle distance? And much more!
We will teach you how to calculate the great circle distance between two points, both on a sphere and on a more realistic shape for Earth. What are you waiting for? Get ready for departure!
What is the great circle? Introduction to the great circle between two points
We can give you two definitions of the great circle, one of them complex, albeit straightforward; the other much more intuitive. Before the definitions, though, it's better to introduce the situations in which we test our great circle calculator.
We define great circles on solid shapes with some degree of rotational symmetry, namely spheres and ellipsoids. On such figures, the rules of the geometry we meet every day are somehow… bent. Let's see how!
What is a great circle: a mathematical definition
The great circle is the closed circle deriving from the intersection of a sphere and a plane passing through the center of the sphere. The intersection traces the largest possible closed path on a sphere.
This definition, of course, allows you to create infinitely many great circles. The same holds if we consider the great circle passing through a single point on the sphere's surface.
This infinite multiplicity is eventually broken when we try to find the great circle between two points. In fact, by choosing two distinct points on a sphere, you can trace a single great circle between them.
🙋 There is an exception to this rule! If the two points are antipodes (exactly opposite each other on the globe), you can again find infinitely many great circles passing through them.
When considering two points and the great circle between them, we can define (and calculate) the distance between them along the great circle. There are two possible values:
- the longer path; and
- the shorter path.
The shorter path leads us to the second definition of the great circle.
The great circle definition based on the distance between two points
The great circle is the circle containing the shortest possible path between two points on a sphere. This is exactly why long-haul flights follow curved routes on a flat map: the straight line you draw on the map is not the shortest way to travel across a curved planet.
So, what is the great circle distance between two points? And why is it important?
The great circle distance is the length of the shortest arc joining two points on the surface of a sphere. Because Earth is (almost) a sphere, this is the closest thing we have to a "straight line" for ships and aircraft. Airlines plan routes along great circles to save fuel and time, which is why a flight from Los Angeles to London arcs up toward Greenland instead of heading in what looks like a straight easterly line on a Mercator map.
The great circle distance is also known as the orthodromic distance. It differs from the rhumb line (loxodrome) distance, which follows a constant compass bearing but is longer.
How do I calculate the great circle distance?
The most numerically stable way to compute the great circle distance is the haversine formula. Given two points with latitudes φ₁, φ₂ and longitudes λ₁, λ₂:
c = 2·atan2(√a, √(1−a))
d = R · c
where:
- Δφ = φ₂ − φ₁ is the difference in latitude;
- Δλ = λ₂ − λ₁ is the difference in longitude;
- c is the central angle (in radians) between the two points; and
- R is the radius of the Earth (mean radius R = 6371.0088 km).
For a more realistic Earth, this calculator also offers the ellipsoidal (WGS-84) model, which uses Vincenty's inverse formula on a flattened ellipsoid. It is a fraction of a percent more accurate over long distances because Earth is slightly wider at the equator than pole-to-pole.
How many great circles are there?
On any sphere there are infinitely many great circles — the equator and every line of longitude (meridian) are examples. Lines of latitude other than the equator are not great circles, because their planes do not pass through the center of the sphere; they are called small circles.
Examples of great circle distance calculation
Let's use the calculator's default cities:
- Los Angeles: 34.052235° N, 118.243683° W
- London: 51.507351° N, 0.127758° W
The initial bearing from Los Angeles is roughly north-east, which is why the flight path curves up over Canada and the tip of Greenland — that arc really is the shortest way across the globe.
How to use the great circle calculator
- Pick a distance model — spherical (haversine) for the classic great-circle distance, or ellipsoidal (WGS-84) for the geodesic on a more realistic Earth.
- Choose your units — metric (kilometers) or US/Imperial (miles). Results also include nautical miles and meters.
- Enter the coordinates of both points as decimal degrees. Use negative values for the southern and western hemispheres.
- Read the distance, the central angle, the initial and final bearings, and the midpoint of the route.
Unit Systems
Metric System
Distances are shown primarily in kilometers (km), with meters, miles, and nautical miles also provided.
US / Imperial System
Distances are shown primarily in miles (mi), with kilometers, nautical miles, and meters also provided. Nautical miles are the standard unit in aviation and marine navigation (1 nmi = 1.852 km).
FAQs
Why do flights follow curved paths instead of straight lines?
On a flat (Mercator) map, the shortest route across a curved planet looks curved. The great circle is the true shortest path, so aircraft follow it — even though the map makes it look like a detour.
Is the great circle distance the same as the straight-line distance?
No. The straight-line (chord) distance tunnels through the Earth. The great circle distance is measured along the surface, which is what you actually travel.
What is the difference between a great circle and a rhumb line?
A great circle is the shortest path but requires continuously changing your compass heading. A rhumb line keeps a constant bearing, which is easier to steer but always longer (except along the equator or a meridian).
What happens with antipodal points?
If the two points are exactly opposite each other on the globe, there are infinitely many great circles of equal length between them, and the ellipsoidal (Vincenty) formula may not converge. The spherical model still returns a distance of half the Earth's circumference.
Which model should I use, spherical or ellipsoidal?
The spherical (haversine) model is fast and accurate to about 0.5%. The ellipsoidal (WGS-84) model is the best choice when you need sub-meter precision over long distances, since it accounts for Earth's flattening.