# Length of the direct common tangent between two externally touching circles

Given two circles, of given radii, that touch each other externally. The task is to find the length of the direct common tangent between the circles.**Examples:**

Input:r1 = 5, r2 = 9Output:13.4164Input:r1 = 11, r2 = 13Output:23.9165

**Approach**

- Let the radii be
**r1**&**r2**respectively. - Draw a line
**OR**parallel to**PQ** **angle OPQ = 90 deg****angle Oâ€™QP = 90 deg**

{ line joining the centre of the circle to the point of contact makes an angle of 90 degrees with the tangent }**angle OPQ + angle Oâ€™QP = 180****OP || QR**- Since opposite sides are parallel and interior angles are 90, therefore
**OPQR**is a rectangle. - So
**OP = QR = r1**and**PQ = OR = r1+r2** - In triangle
**OO’R****angle ORO’ = 90**

By**Pythagoras theorem****OR^2 + O’R^2 = OO’^2****OO’^2 = (r1+r2)^2 + (r1-r2)^2** - So,
**OO’ = 2âˆš(r1*r2)**

Below is the implementation of the above approach:

## C++

`// C++ program to find the length of the direct` `// common tangent between two circles` `// which externally touch each other` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to find the length` `// of the direct common tangent` `void` `lengtang(` `double` `r1, ` `double` `r2)` `{` ` ` `cout << ` `"The length of the "` ` ` `<< ` `"direct common tangent is "` ` ` `<< 2 * ` `sqrt` `(r1 * r2) << endl;` `}` `// Driver code` `int` `main()` `{` ` ` `double` `r1 = 5, r2 = 9;` ` ` `lengtang(r1, r2);` ` ` `return` `0;` `}` |

## Java

`// Java program to find the length of the direct ` `// common tangent between two circles ` `// which externally touch each other ` `class` `GFG ` `{` ` ` `// Function to find the length ` ` ` `// of the direct common tangent ` ` ` `static` `void` `lengtang(` `double` `r1, ` `double` `r2) ` ` ` `{` ` ` `System.out.println(` `"The length of the "` ` ` `+ ` `"direct common tangent is "` ` ` `+ (` `2` `* Math.sqrt(r1 * r2)));` ` ` `}` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `double` `r1 = ` `5` `, r2 = ` `9` `;` ` ` `lengtang(r1, r2);` ` ` `}` `}` `// This code contributed by Rajput-Ji` |

## Python3

`# Python3 program to find the length ` `# of the direct common tangent ` `# between two circles which ` `# externally touch each other ` `# Function to find the length ` `# of the direct common tangent ` `def` `lengtang(r1, r2):` ` ` `print` `(` `"The length of the direct"` `, ` ` ` `"common tangent is"` `, ` ` ` `2` `*` `(r1 ` `*` `r2)` `*` `*` `(` `1` `/` `2` `)); ` `# Driver code ` `r1 ` `=` `5` `; r2 ` `=` `9` `; ` `lengtang(r1, r2); ` `# This code contributed ` `# by PrinciRaj1992 ` |

## C#

`// C# program to find the length of the direct ` `// common tangent between two circles ` `// which externally touch each other` `using` `System;` `class` `GFG` `{` ` ` `// Function to find the length ` ` ` `// of the direct common tangent ` ` ` `static` `void` `lengtang(` `double` `r1, ` `double` `r2) ` ` ` `{` ` ` `Console.WriteLine(` `"The length of the "` ` ` `+ ` `"direct common tangent is "` ` ` `+ (2 * Math.Sqrt(r1 * r2)));` ` ` `}` ` ` ` ` `// Driver code ` ` ` `static` `public` `void` `Main ()` ` ` `{` ` ` `double` `r1 = 5, r2 = 9;` ` ` `lengtang(r1, r2);` ` ` `}` `}` `// This code contributed by ajit.` |

## PHP

`<?php` `// PHP program to find the length of the direct ` `// common tangent between two circles ` `// which externally touch each other ` `// Function to find the length ` `// of the direct common tangent ` `function` `lengtang(` `$r1` `, ` `$r2` `) ` `{ ` ` ` `echo` `"The length of the "` ` ` `, ` `"direct common tangent is "` ` ` `, 2 * sqrt(` `$r1` `* ` `$r2` `) ; ` `} ` `// Driver code ` `$r1` `= 5; ` `$r2` `= 9; ` `lengtang(` `$r1` `, ` `$r2` `); ` `// This code is contributed by AnkitRai01` `?>` |

## Javascript

`<script>` `// javascript program to find the length of the direct ` `// common tangent between two circles ` `// which externally touch each other ` `// Function to find the length ` `// of the direct common tangent ` `function` `lengtang(r1 , r2) ` `{` ` ` `document.write(` `"The length of the "` ` ` `+ ` `"direct common tangent is "` ` ` `+ (2 * Math.sqrt(r1 * r2)).toFixed(5));` `}` `// Driver code ` `var` `r1 = 5, r2 = 9;` `lengtang(r1, r2);` `// This code contributed by Princi Singh ` `</script>` |

**Output:**

The length of the direct common tangent is 13.4164

**Time Complexity:** O(log(n)) because using inbuilt sqrt function

**Auxiliary Space: ** O(1)