# Number of triangles formed by joining vertices of n-sided polygon with one side common

Given **N**-sided polygon we need to find the number of triangles formed by joining the vertices of the given polygon with exactly one side being common.**Examples:**

Input :6Output :12

The image below is of a triangle forming inside a Hexagon by joining vertices as shown above. The two triangles formed has one side (AB) common with that of a polygon.It depicts that with one edge of a hexagon we can make two triangles with one side common. We can’t take C or F as our third vertex as it will make 2 sides common with the hexagon.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the

DSA Self Paced Courseat a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please referComplete Interview Preparation Course.In case you wish to attend

live classeswith experts, please referDSA Live Classes for Working ProfessionalsandCompetitive Programming Live for Students.No of triangles formed ie equal to 12 as there are 6 edges in a hexagon.

Input :5Output :5

**Approach :**

- To make a triangle with one side common with a polygon the two vertices adjacent to the chosen common vertices cannot be considered as the third vertex of a triangle.
- First select any one edge from the polygon. Consider this edge to be the common edge. Number of ways to select an edge in a polygon would be equal to n.
- Now ,to form a triangle ,select any of the (n-4) vertices left .Two vertices of the common edge and two vertices adjacent to the common edge cannot be considered.
- Number of triangle formed by joining the vertices of an n-sided polygon with one side common would be equal to n * ( n – 4) .

Below is the implementation of the above approach:

## C++

`// C++ program to implement` `// the above problem` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to find the number of triangles` `void` `findTriangles(` `int` `n)` `{` ` ` `int` `num;` ` ` `num = n * (n - 4);` ` ` `// print the number of triangles` ` ` `cout << num;` `}` `// Driver code` `int` `main()` `{` ` ` `// initialize the number of sides of a polygon` ` ` `int` `n;` ` ` `n = 6;` ` ` `findTriangles(n);` ` ` `return` `0;` `}` |

## Java

`// Java program to implement` `// the above problem` `class` `GFG` `{` ` ` `// Function to find the number of triangles` `static` `void` `findTriangles(` `int` `n)` `{` ` ` `int` `num;` ` ` `num = n * (n - ` `4` `);` ` ` `// print the number of triangles` ` ` `System.out.println(num);` `}` `// Driver code` `public` `static` `void` `main(String [] args)` `{` ` ` `// initialize the number of sides of a polygon` ` ` `int` `n;` ` ` `n = ` `6` `;` ` ` `findTriangles(n);` `}` `}` `// This code is contributed by ihritik` |

## Python3

`# Python3 program to implement` `# the above problem` `# Function to find the number of triangles` `def` `findTriangles(n):` ` ` `num ` `=` `n ` `*` `(n ` `-` `4` `)` ` ` `# print the number of triangles` ` ` `print` `(num)` `# Driver code` `# initialize the number of sides of a polygon` `n ` `=` `6` `findTriangles(n)` `# This codee is contributed by mohit kumar 29` |

## C#

`// C# program to implement` `// the above problem` `using` `System;` `class` `GFG` `{` ` ` `// Function to find the number of triangles` `static` `void` `findTriangles(` `int` `n)` `{` ` ` `int` `num;` ` ` `num = n * (n - 4);` ` ` `// print the number of triangles` ` ` `Console.WriteLine(num);` `}` `// Driver code` `public` `static` `void` `Main()` `{` ` ` `// initialize the number of sides of a polygon` ` ` `int` `n;` ` ` `n = 6;` ` ` `findTriangles(n);` `}` `}` `// This code is contributed by ihritik` |

## Javascript

`<script>` `// JavaScript program to implement` `// the above problem` ` ` `// Function to find the number of triangles` `function` `findTriangles(n)` `{` ` ` `var` `num;` ` ` `num = n * (n - 4);` ` ` `// print the number of triangles` ` ` `document.write(num);` `}` `// Driver code` `// initialize the number of sides of a polygon` `var` `n;` `n = 6;` `findTriangles(n);` `// This code contributed by shikhasingrajput` `</script>` |

**Output:**

12

**Time Complexity: **O(1)