The earth is round, not flat.

Euclidean distances work for flat surfaces but not curved surfaces.

It is often needed to calculate (approximate) the shortest distance between two points on the surface of the earth.

A location in geographic coordinates is expressed as a longitude and a latitude. Each are angles.

There are many ways to calculate the shortest distance between two points on the surface of the earth.

Here the assumption is made that the earth is a perfect sphere.

The spherical law of cosines can be used to approximate the shortest distance between two points on the surface of the earth (for small, not large, distances).

Note that coordinates on the Internet can vary for places depending on where in that location the coordinates are taken.

Actual distances by roads, of course, would be longer.

Given 2 points on the surface of the earth in degrees.

• Point 1: (lat1d, long1d)
• Point 2: (lat2d, long2d)

The radius of the earth is about 3958.8 miles.

Convert all latitudes and longitudes from degrees to radians using the following relationships

• 2π radians = 360 degrees
• π radians = 180 degrees

This relationship can be used to convert, say, lat1d (latitude in degrees) to lat1r (latitude in radians), etc.

The symbol π is pi and is approximated by 3.1415.... In an actual programming language, one would use a constant for pi that approximates pi to the precision of that system. The distance d between the two points is then as follows, where acos is the arccos function (i.e., inverse cos) and all angles are in radians.

If radius is in miles, the distance is in miles. If radius is in kilometers, then the distance is in kilometers.

A more mathematical formula is as follows.


Where:

• R is radius
• φ₁ is latr1r
• φ₂ is latr2r
• θ₁ is long1r
• θ₂ is long2r