How do you calculate the distance between to points on the earth?

Between any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. Between two points which are directly opposite each other (called antipodal points) there are infinitely many great circles, but all have the same length, equal to half the circumference of the circle, or πr, where r is the radius of the sphere.