Great-circle distance

Great circle distance is the shortest distance between any two points on a sphere.

Because the Earth is approximately spherical, the great circle distance is commonly used to find the distance (e.g. in miles) between two coordinates (latitude/longitude pairs) on a map. First, some definitions:

Definitions

R

is the radius of the Earth (

\approx 6367000

meters)

\delta

is latitude

\lambda

is longitude

The Formula

gc(\delta ,\lambda ,\delta ',\lambda ')=

2\cdot R\cdot \arcsin {(\min {(1,{\sqrt {(\sin {({\frac {\delta '-\delta }{2}})})^{2}+\cos {\delta }\cdot \cos {\delta '}\cdot (\sin {({\frac {\lambda '-\lambda }{2}})})^{2}}})})}

Practical Application

Choose two points

In order to use this formula for anything practical you will need two sets of coordinates. I'll provide the latitude and longitude of two airports:

Nashville International Airport (BNA) in Nashville, TN, USA: W 86°40.2, N 36°7.2'
Los Angeles International Airport (LAX) in Los Angeles, CA, USA: W 118°24.0, N 33°56.4'

Convert coordinates

You will have to convert these coordinates to a more mathematically friendly form using a simple method before you can use them effectively in a formula. After conversion, the coordinates become:

BNA: $-86.670000(\delta ),36.120000(\lambda )$
LAX: $-118.400000(\delta '),33.940000(\lambda ')$