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.
Input : 6
Output : 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.
No of triangles formed ie equal to 12 as there are 6 edges in a hexagon.
Input : 5
Output : 5
- 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:
Time Complexity: O(1)
Auxiliary Space: O(1)