The great circle distance is the length of an arc between two points belonging to a circumference passing through the center of a sphere or ellipsoid.

Great circles are straight lines on a sphere but bend when you represent the shape in two dimensions. That's why aerial routes, straight in real life, look curved on a map.

To calculate the great circle distance on a sphere, take the points A = (φA, λA) and B = (φB, λB), and follow the steps:

1. Compute the product of the cosines of the latitudes and the difference of longitudes:
   t1 = cos(φA) × cos(φB) × cos(λA - λB)
2. Compute the product of the sines of the longitudes:
   t2 = sin(λA) × sin(λB)
3. Calculate the inverse cosine of the sum of the two quantities, and multiply it by the radius:
   d = r × t1+ t2

This is the great circle distance.

If you consider Earth to be a sphere with a radius r = 6371 km, then the great circles have all the same length, equal to 40,030 km.

Earth, however, is an ellipsoid, and the great circle lengths vary between two values:

• The circumference passing for the poles: 40,007 km; and
• The equatorial circumference: 40,075 km.