Haversine formula

The hav­er­sine formula is used to cal­cu­late the distance between two points on the Earth’s surface spe­cified in lon­gitude and latitude.

d is the distance between two points with lon­gitude and latitude (ψ,φ) and r is the radius of the Earth.

As an example I have cal­cu­lated the distance between Fermilab in Illinois (41° 49′ 55″ N, 88° 15′ 26″ W) and CERN’s Meyrin campus in Switzer­land (46° 14′ 3″ N, 6° 3′ 10″ E). There’s a little too much maths for this site to handle so I have included a .PDF file of the working below.

The value cal­cu­lated is 7084 km, which isn’t quite correct. This is because the formula assumes that the Earth is a perfect sphere when in fact it is an oblate spheroid. To com­pensate for this Vincenty’s Formulae must be used; these are much more com­plic­ated but give a more accurate value of 7103 km.