# Haversine formula to find distance between two points on a sphere

The **Haversine** formula calculates the shortest distance between two points on a sphere using their latitudes and longitudes measured along the surface. It is important for use in navigation. The haversine can be expressed in trignometric function as:

The haversine of the central angle (which is d/r) is calculated by the following formula:

where r is the radius of earth(6371 km), d is the distance between two points, is latitude of the two points and is longitude of the two points respectively.

Solving d by applying the inverse haversine or by using the inverse sine function, we get:

or

The distance between Big Ben in London (51.5007° N, 0.1246° W) and The Statue of Liberty in

New York (40.6892° N, 74.0445° W) is 5574.8 km. This is not the exact measurement because the

formula assumes that the Earth is a perfect sphere when in fact it is an oblate spheroid.

Below is the implementation of the above formulae:

## C++

`// C++ program for the haversine formula ` `// C++ program for the ` `// haversine formula ` `#include <iostream> ` `#include <cmath> ` `using` `namespace` `std; ` ` ` `static` `double` `haversine(` `double` `lat1, ` `double` `lon1, ` ` ` `double` `lat2, ` `double` `lon2) ` ` ` `{ ` ` ` `// distance between latitudes ` ` ` `// and longitudes ` ` ` `double` `dLat = (lat2 - lat1) * ` ` ` `M_PI / 180.0; ` ` ` `double` `dLon = (lon2 - lon1) * ` ` ` `M_PI / 180.0; ` ` ` ` ` `// convert to radians ` ` ` `lat1 = (lat1) * M_PI / 180.0; ` ` ` `lat2 = (lat2) * M_PI / 180.0; ` ` ` ` ` `// apply formulae ` ` ` `double` `a = ` `pow` `(` `sin` `(dLat / 2), 2) + ` ` ` `pow` `(` `sin` `(dLon / 2), 2) * ` ` ` `cos` `(lat1) * ` `cos` `(lat2); ` ` ` `double` `rad = 6371; ` ` ` `double` `c = 2 * ` `asin` `(` `sqrt` `(a)); ` ` ` `return` `rad * c; ` ` ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `double` `lat1 = 51.5007; ` ` ` `double` `lon1 = 0.1246; ` ` ` `double` `lat2 = 40.6892; ` ` ` `double` `lon2 = 74.0445; ` ` ` ` ` `cout << haversine(lat1, lon1, ` ` ` `lat2, lon2) << ` `" K.M."` `; ` ` ` `return` `0; ` `} ` ` ` `// This code is contributed ` `// by Mahadev. ` |

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## Java

`// Java program for the haversine formula ` `public` `class` `Haversine { ` ` ` ` ` `static` `double` `haversine(` `double` `lat1, ` `double` `lon1, ` ` ` `double` `lat2, ` `double` `lon2) ` ` ` `{ ` ` ` `// distance between latitudes and longitudes ` ` ` `double` `dLat = Math.toRadians(lat2 - lat1); ` ` ` `double` `dLon = Math.toRadians(lon2 - lon1); ` ` ` ` ` `// convert to radians ` ` ` `lat1 = Math.toRadians(lat1); ` ` ` `lat2 = Math.toRadians(lat2); ` ` ` ` ` `// apply formulae ` ` ` `double` `a = Math.pow(Math.sin(dLat / ` `2` `), ` `2` `) + ` ` ` `Math.pow(Math.sin(dLon / ` `2` `), ` `2` `) * ` ` ` `Math.cos(lat1) * ` ` ` `Math.cos(lat2); ` ` ` `double` `rad = ` `6371` `; ` ` ` `double` `c = ` `2` `* Math.asin(Math.sqrt(a)); ` ` ` `return` `rad * c; ` ` ` `} ` ` ` ` ` `// Driver Code ` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{ ` ` ` `double` `lat1 = ` `51.5007` `; ` ` ` `double` `lon1 = ` `0.1246` `; ` ` ` `double` `lat2 = ` `40.6892` `; ` ` ` `double` `lon2 = ` `74.0445` `; ` ` ` `System.out.println(haversine(lat1, lon1, lat2, lon2) + ` `" K.M."` `); ` ` ` `} ` `} ` |

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## Python 3

`# Python 3 program for the ` `# haversine formula ` `import` `math ` ` ` `# Python 3 program for the ` `# haversine formula ` `def` `haversine(lat1, lon1, lat2, lon2): ` ` ` ` ` `# distance between latitudes ` ` ` `# and longitudes ` ` ` `dLat ` `=` `(lat2 ` `-` `lat1) ` `*` `math.pi ` `/` `180.0` ` ` `dLon ` `=` `(lon2 ` `-` `lon1) ` `*` `math.pi ` `/` `180.0` ` ` ` ` `# convert to radians ` ` ` `lat1 ` `=` `(lat1) ` `*` `math.pi ` `/` `180.0` ` ` `lat2 ` `=` `(lat2) ` `*` `math.pi ` `/` `180.0` ` ` ` ` `# apply formulae ` ` ` `a ` `=` `(` `pow` `(math.sin(dLat ` `/` `2` `), ` `2` `) ` `+` ` ` `pow` `(math.sin(dLon ` `/` `2` `), ` `2` `) ` `*` ` ` `math.cos(lat1) ` `*` `math.cos(lat2)); ` ` ` `rad ` `=` `6371` ` ` `c ` `=` `2` `*` `math.asin(math.sqrt(a)) ` ` ` `return` `rad ` `*` `c ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` `lat1 ` `=` `51.5007` ` ` `lon1 ` `=` `0.1246` ` ` `lat2 ` `=` `40.6892` ` ` `lon2 ` `=` `74.0445` ` ` ` ` `print` `(haversine(lat1, lon1,lat2, lon2), ` `"K.M."` `) ` ` ` `# This code is contributed ` `# by ChitraNayal ` |

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## C#

`// C# program for the haversine formula ` `using` `System; ` `class` `GFG ` `{ ` ` ` `static` `double` `haversine(` `double` `lat1, ` `double` `lon1, ` ` ` `double` `lat2, ` `double` `lon2) ` `{ ` ` ` `// distance between latitudes and longitudes ` ` ` `double` `dLat = (Math.PI / 180) * (lat2 - lat1); ` ` ` `double` `dLon = (Math.PI / 180) * (lon2 - lon1); ` ` ` ` ` `// convert to radians ` ` ` `lat1 = (Math.PI / 180) * (lat1); ` ` ` `lat2 = (Math.PI / 180) * (lat2); ` ` ` ` ` `// apply formulae ` ` ` `double` `a = Math.Pow(Math.Sin(dLat / 2), 2) + ` ` ` `Math.Pow(Math.Sin(dLon / 2), 2) * ` ` ` `Math.Cos(lat1) * Math.Cos(lat2); ` ` ` `double` `rad = 6371; ` ` ` `double` `c = 2 * Math.Asin(Math.Sqrt(a)); ` ` ` `return` `rad * c; ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main() ` `{ ` ` ` `double` `lat1 = 51.5007; ` ` ` `double` `lon1 = 0.1246; ` ` ` `double` `lat2 = 40.6892; ` ` ` `double` `lon2 = 74.0445; ` ` ` `Console.WriteLine(haversine(lat1, lon1, ` ` ` `lat2, lon2) + ` `" K.M."` `); ` `} ` `} ` ` ` `// This code is contributed ` `// by Akanksha Rai(Abby_akku) ` |

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## PHP

`<?php ` `// PHP program for the ` `// haversine formula ` ` ` `function` `haversine(` `$lat1` `, ` `$lon1` `, ` ` ` `$lat2` `, ` `$lon2` `) ` `{ ` ` ` `// distance between latitudes ` ` ` `// and longitudes ` ` ` `$dLat` `= (` `$lat2` `- ` `$lat1` `) * ` ` ` `M_PI / 180.0; ` ` ` `$dLon` `= (` `$lon2` `- ` `$lon1` `) * ` ` ` `M_PI / 180.0; ` ` ` ` ` `// convert to radians ` ` ` `$lat1` `= (` `$lat1` `) * M_PI / 180.0; ` ` ` `$lat2` `= (` `$lat2` `) * M_PI / 180.0; ` ` ` ` ` `// apply formulae ` ` ` `$a` `= pow(sin(` `$dLat` `/ 2), 2) + ` ` ` `pow(sin(` `$dLon` `/ 2), 2) * ` ` ` `cos` `(` `$lat1` `) * ` `cos` `(` `$lat2` `); ` ` ` `$rad` `= 6371; ` ` ` `$c` `= 2 * asin(sqrt(` `$a` `)); ` ` ` `return` `$rad` `* ` `$c` `; ` `} ` ` ` `// Driver code ` `$lat1` `= 51.5007; ` `$lon1` `= 0.1246; ` `$lat2` `= 40.6892; ` `$lon2` `= 74.0445; ` ` ` `echo` `haversine(` `$lat1` `, ` `$lon1` `, ` ` ` `$lat2` `, ` `$lon2` `) . ` ` ` `" K.M."` `; ` ` ` `// This code is contributed ` `// by Akanksha Rai(Abby_akku) ` `?> ` |

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**Output:**

5574.840456848555 K.M.

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