# Total number of triangles formed when there are H horizontal and V vertical lines

Given a triangle ABC. H horizontal lines from side AB to AC (as shown in fig.) and V vertical lines from vertex A to side BC are drawn, the task is to find the total no. of triangles formed.

Examples:

Input: H = 2, V = 2
Output: 18

As we see in the image above, total triangles formed are 18.

Input: H = 3, V = 4
Output: 60

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: As we see in the images below, we can derive a general formula for above problem:

1. If there are only h horizontal lines then total triangles are (h + 1).
2. If there are only v vertical lines then total triangles are (v + 1) * (v + 2) / 2..
3. So, total triangles are Triangles formed by horizontal lines * Triangles formed by vertical lines i.e. (h + 1) * (( v + 1) * (v + 2) / 2).

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` `#define LLI long long int ` ` `  `// Function to return total triangles ` `LLI totalTriangles(LLI h, LLI v) ` `{ ` `    ``// Only possible triangle is ` `    ``// the given triangle ` `    ``if` `(h == 0 && v == 0) ` `        ``return` `1; ` ` `  `    ``// If only vertical lines are present ` `    ``if` `(h == 0) ` `        ``return` `((v + 1) * (v + 2) / 2); ` ` `  `    ``// If only horizontal lines are present ` `    ``if` `(v == 0) ` `        ``return` `(h + 1); ` ` `  `    ``// Return total triangles ` `    ``LLI Total = (h + 1) * ((v + 1) * (v + 2) / 2); ` ` `  `    ``return` `Total; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `h = 2, v = 2; ` `    ``cout << totalTriangles(h, v); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach ` `class` `GFG { ` ` `  `    ``// Function to return total triangles ` `    ``public` `static` `int` `totalTriangles(``int` `h, ``int` `v) ` `    ``{ ` `        ``// Only possible triangle is ` `        ``// the given triangle ` `        ``if` `(h == ``0` `&& v == ``0``) ` `            ``return` `1``; ` ` `  `        ``// If only vertical lines are present ` `        ``if` `(h == ``0``) ` `            ``return` `((v + ``1``) * (v + ``2``) / ``2``); ` ` `  `        ``// If only horizontal lines are present ` `        ``if` `(v == ``0``) ` `            ``return` `(h + ``1``); ` ` `  `        ``// Return total triangles ` `        ``int` `total = (h + ``1``) * ((v + ``1``) * (v + ``2``) / ``2``); ` ` `  `        ``return` `total; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `h = ``2``, v = ``2``; ` `        ``System.out.print(totalTriangles(h, v)); ` `    ``} ` `} `

## C#

 `// C# implementation of the approach  ` `using` `System; ` ` `  `class` `GFG  ` `{  ` ` `  `    ``// Function to return total triangles  ` `    ``public` `static` `int` `totalTriangles(``int` `h, ``int` `v)  ` `    ``{  ` `        ``// Only possible triangle is  ` `        ``// the given triangle  ` `        ``if` `(h == 0 && v == 0)  ` `            ``return` `1;  ` ` `  `        ``// If only vertical lines are present  ` `        ``if` `(h == 0)  ` `            ``return` `((v + 1) * (v + 2) / 2);  ` ` `  `        ``// If only horizontal lines are present  ` `        ``if` `(v == 0)  ` `            ``return` `(h + 1);  ` ` `  `        ``// Return total triangles  ` `        ``int` `total = (h + 1) * ((v + 1) * (v + 2) / 2);  ` ` `  `        ``return` `total;  ` `    ``}  ` ` `  `    ``// Driver code  ` `    ``public` `static` `void` `Main()  ` `    ``{  ` `        ``int` `h = 2, v = 2;  ` `        ``Console.Write(totalTriangles(h, v));  ` `    ``}  ` `}  ` ` `  `// This code is contributed by Ryuga `

## Python3

 `# Python3 implementation of the approach ` ` `  `# Function to return total triangles ` `def` `totalTriangles(h, v): ` `     `  `    ``# Only possible triangle is  ` `    ``# the given triangle ` `    ``if` `(h ``=``=` `0` `and` `v ``=``=` `0``): ` `        ``return` `1` ` `  `    ``# If only vertical lines are present ` `    ``if` `(h ``=``=` `0``): ` `        ``return` `((v ``+` `1``) ``*` `(v ``+` `2``) ``/` `2``) ` ` `  `    ``# If only horizontal lines are present ` `    ``if` `(v ``=``=` `0``): ` `        ``return` `(h ``+` `1``) ` ` `  `    ``# Return total triangles ` `    ``total ``=` `(h ``+` `1``) ``*` `((v ``+` `1``) ``*` `(v ``+` `2``) ``/` `2``) ` ` `  `    ``return` `total ` ` `  `# Driver code ` `h ``=` `2` `v ``=` `2` `print``(``int``(totalTriangles(h, v))) `

## PHP

 ` `

Output:

```18
```

Time Complexity: O(1)
Auxiliary Space: O(1)

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