Number of triangles formed by joining vertices of n-sided polygon with two common sides and no common sides

31 May, 2022
Given N-sided polygon we need to find the total number of triangles formed by joining the vertices of the given polygon with exactly two sides being common and no side being common.Examples:  
Input : N = 6 Output : 6 2 The image below is of a triangle forming inside a Hexagon by joining vertices as shown above. The triangle formed has two sides (AB and BC) common with that of a polygon. Similarly BC and CD can make one triangle. With this, we can say that there will be a total of 6 triangles possible having two sides common with that of a polygon. The second image of a hexagon, a triangle is formed with no side common with that of a polygon. There will be just 2 triangles possible, BFD and ACE.  
Triangle with two side common and no side common of the Hexagon
Number of triangles formed are 6 and 2 with two side common and with no side common respectively.Input : N = 7 Output : 7 7  
 
Approach :  
To make a triangle two side common with a polygon we will take any one side of a n-sided polygon, take one vertex of the chosen side and join an edge adjacent to the vertex of the other vertex.
Traversing through each vertex and adjoining an edge adjacent to the vertex of the other vertex ,there will be N number of triangles having two side common.
Now, to calculate the number of triangles with no side common subtract the total number of triangles with one side common and the total number of triangles with two side from the total number of triangles possible in a polygon.
Triangles with no common side = Total triangles ( nC3 ) – one side common triangles ( n * ( n – 4 ) – two side common triangles ( n ).
Thus number of triangles with no common side with the polygon would be equal to n * ( n – 4 ) * ( n – 5 ) / 6.
Note:To calculate the number of triangles having one side common with that of a polygon click hereBelow is the implementation of the above approach:  
C++
Java
Python3
C#
Javascript
// C++ program to implement// the above problem#include <bits/stdc++.h>using namespace std; // Function to find the number of trianglesvoid findTriangles(int n){    int num = n;     // print the number of triangles    // having two side common    cout << num << " ";     // print the number of triangles    // having no side common    cout << num * (num - 4) * (num - 5) / 6;} // Driver codeint main(){    // initialize the number    // of sides of a polygon    int n;    n = 6;     findTriangles(n);     return 0;}
// Java program to implement// the above problemimport java.io.*; class GFG{  // Function to find the number of trianglesstatic void findTriangles(int n){    int num = n;     // print the number of triangles    // having two side common    System.out.print( num + " ");     // print the number of triangles    // having no side common    System.out.print( num * (num - 4) * (num - 5) / 6);} // Driver codepublic static void main (String[] args){    // initialize the number    // of sides of a polygon    int n;    n = 6;     findTriangles(n);}} // This code is contributed by anuj_67..
# Python3 program to implement# the above problem # Function to find the number of trianglesdef findTriangles(n):    num = n          # print the number of triangles    # having two side common    print(num, end = " ")     # print the number of triangles    # having no side common    print(num * (num - 4) * (num - 5) // 6) # Driver code # initialize the number# of sides of a polygonn = 6; findTriangles(n) # This code is contributed by Mohit Kumar
// C# program to implement// the above problemusing System; class GFG{  // Function to find the number of trianglesstatic void findTriangles(int n){    int num = n;     // print the number of triangles    // having two side common    Console.Write( num + " ");     // print the number of triangles    // having no side common    Console.WriteLine( num * (num - 4) * (num - 5) / 6);} // Driver codepublic static void Main (){    // initialize the number    // of sides of a polygon    int n;    n = 6;     findTriangles(n);}} // This code is contributed by anuj_67..
<script> // javascript program to implement// the above problem // Function to find the number of triangles function findTriangles(n){    var num = n;     // print the number of triangles    // having two side common    document.write( num + " ");     // print the number of triangles    // having no side common    document.write( num * (num - 4) * (num - 5) / 6);} // Driver code// initialize the number// of sides of a polygonvar n;n = 6; findTriangles(n);  // This code is contributed by 29AjayKumar </script>
6 2
 
Time Complexity: O(1)
Auxiliary Space: O(1) 
mohit kumar 29
vt_m
nidhi_biet
29AjayKumar
subhamkumarm348
Geometric
Mathematical
Mathematical
Geometric
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.