Given two circles, of given radii, have there centres a given distance apart, such that the circles intersect each other at two points. The task is to find the length of the direct common tangent between the circles.

**Examples:**

Input:r1 = 4, r2 = 6, d = 3Output:2.23607Input:r1 = 14, r2 = 43, d = 35Output:19.5959

**Approach**:

**r1**&

**r2**respectively.

**d**units.

**angle OPQ = 90 deg**

angle O’QP = 90 deg

angle O’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 O’QP = 180 deg**

OP || QR

OP || QR

**OP = QR = r1 and PQ = OR = d**

**triangle OO’R**

**angle ORO’ = 90**

By **Pythagoras theorem**

**OR^2 + O’R^2 = (OO’^2)
OR^2 + (r1-r2)^2 = d^2**

**OR^2= d^2-(r1-r2)^2**

OR = √{d^2-(r1-r2)^2}

OR = √{d^2-(r1-r2)^2}

Below is the implementation of the above approach:

## C++

`// C++ program to find ` `// the length of the direct ` `// common tangent between two circles ` `// which intersect 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, ` `double` `d) ` `{ ` ` ` `cout << ` `"The length of the direct"` ` ` `<<` `" common tangent is "` ` ` `<< ` `sqrt` `(` `pow` `(d, 2) - ` `pow` `((r1 - r2), 2)) ` ` ` `<< endl; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `double` `r1 = 4, r2 = 6, d = 3; ` ` ` `lengtang(r1, r2, d); ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java program to find ` `// the length of the direct ` `// common tangent between two circles ` `// which intersect each other ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the length of ` ` ` `// the direct common tangent ` ` ` `static` `void` `lengtang(` `double` `r1, ` `double` `r2, ` `double` `d) ` ` ` `{ ` ` ` `System.out.println(` `"The length of the direct"` ` ` `+ ` `" common tangent is "` ` ` `+ (Math.sqrt(Math.pow(d, ` `2` `) - ` ` ` `Math.pow((r1 - r2), ` `2` `)))); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{ ` ` ` `double` `r1 = ` `4` `, r2 = ` `6` `, d = ` `3` `; ` ` ` `lengtang(r1, r2, d); ` ` ` `} ` `} ` ` ` `/* This code contributed by PrinciRaj1992 */` |

*chevron_right*

*filter_none*

## Python3

`# Python program to find ` `# the length of the direct ` `# common tangent between two circles ` `# which intersect each other ` ` ` `# Function to find the length of ` `# the direct common tangent ` `def` `lengtang(r1, r2, d): ` ` ` `print` `(` `"The length of the direct common tangent is "` ` ` `,((d` `*` `*` `2` `) ` `-` `((r1 ` `-` `r2)` `*` `*` `2` `))` `*` `*` `(` `1` `/` `2` `)); ` ` ` ` ` `# Driver code ` `r1 ` `=` `4` `; r2 ` `=` `6` `; d ` `=` `3` `; ` `lengtang(r1, r2, d); ` ` ` `# This code has been contributed by 29AjayKumar ` |

*chevron_right*

*filter_none*

## C#

`// C# program to find ` `// the length of the direct ` `// common tangent between two circles ` `// which intersect each other ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the length of ` ` ` `// the direct common tangent ` ` ` `static` `void` `lengtang(` `double` `r1, ` `double` `r2, ` `double` `d) ` ` ` `{ ` ` ` `Console.WriteLine(` `"The length of the direct"` ` ` `+ ` `" common tangent is "` ` ` `+ (Math.Sqrt(Math.Pow(d, 2) - ` ` ` `Math.Pow((r1 - r2), 2)))); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main(String[] args) ` ` ` `{ ` ` ` `double` `r1 = 4, r2 = 6, d = 3; ` ` ` `lengtang(r1, r2, d); ` ` ` `} ` `} ` ` ` `/* This code contributed by PrinciRaj1992 */` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP program to find ` `// the length of the direct ` `// common tangent between two circles ` `// which intersect each other ` ` ` `// Function to find the length of ` `// the direct common tangent ` `function` `lengtang(` `$r1` `, ` `$r2` `, ` `$d` `) ` `{ ` ` ` `echo` `"The length of the direct common tangent is "` ` ` `,sqrt(pow(` `$d` `, 2) - pow((` `$r1` `- ` `$r2` `), 2)) ; ` `} ` ` ` `// Driver code ` `$r1` `= 4; ` `$r2` `= 6; ` `$d` `= 3; ` `lengtang(` `$r1` `, ` `$r2` `, ` `$d` `); ` ` ` `// This code is contributed by AnkitRai01 ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

The length of the direct common tangent is 2.23607

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.

## Recommended Posts:

- Length of direct common tangent between the two non-intersecting Circles
- Length of the transverse common tangent between the two non intersecting circles
- Length of the direct common tangent between two externally touching circles
- Ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles
- Distance between centers of two intersecting circles if the radii and common chord length is given
- Ratio of the distance between the centers of the circles and the point of intersection of two transverse common tangents to the circles
- Radii of the three tangent circles of equal radius which are inscribed within a circle of given radius
- Radius of the inscribed circle within three tangent circles
- Number of common tangents between two circles if their centers and radius is given
- Length of the perpendicular bisector of the line joining the centers of two circles
- Angle between a chord and a tangent when angle in the alternate segment is given
- Intersecting rectangle when bottom-left and top-right corners of two rectangles are given
- Program to calculate the area between two Concentric Circles
- Length of rope tied around three equal circles touching each other
- Number of triangles formed by joining vertices of n-sided polygon with two common sides and no common sides
- Find Tangent at a given point on the curve
- Check if two given circles touch or intersect each other
- Check whether given circle resides in boundary maintained by two other circles
- Check if a given circle lies completely inside the ring formed by two concentric circles
- Check if two given Circles are Orthogonal or not

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.