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# Length of direct common tangent between the two non-intersecting Circles

• Last Updated : 07 Jun, 2022

Given two circles, of given radii, have there centres a given distance apart, such that the circles don’t touch each other. The task is to find the length of the direct common tangent between the circles.
Examples:

```Input: r1 = 4, r2 = 6, d = 12
Output: 11.8322

Input: r1 = 5, r2 = 9, d = 25
Output: 24.6779```

Approach

• Let the radii of the circles be r1 & r2 respectively.
• Let the distance between the centers be d units.
• Draw a line OR parallel to PQ
• angle OPQ = 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
• Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.
• So OP = QR = r1 and PQ = OR = d
• In triangle OO’R
angle ORO’ = 90
By Pythagoras theorem
OR^2 + O’R^2 = (OO’^2)
OR^2 + (r1-r2)^2 = d^2
• so, OR^2= d^2-(r1-r2)^2
OR = âˆš{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 donot touch each other`   `#include ` `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 = 12;` `    ``lengtang(r1, r2, d);` `    ``return` `0;` `}`

## Java

 `// Java program to find` `// the length of the direct` `// common tangent between two circles` `// which donot touch 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 = ``12``;` `    ``lengtang(r1, r2, d);` `}` `}`   `/* This code contributed by PrinciRaj1992 */`

## Python3

 `# Python3 program to find` `# the length of the direct` `# common tangent between two circles` `# which do not touch each other` `import` `math`   `# 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 ``=` `12``;` `lengtang(r1, r2, d);`   `# This code is contributed by 29AjayKumar`

## C#

 `// C# program to find` `// the length of the direct` `// common tangent between two circles` `// which donot touch 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()` `    ``{` `        ``double` `r1 = 4, r2 = 6, d = 12;` `        ``lengtang(r1, r2, d);` `    ``}` `}`   `// This code is contributed by AnkitRai01`

## PHP

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## Javascript

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Output:

`The length of the direct common tangent is 11.8322`

Time Complexity: O(1)

Auxiliary Space: O(1)

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