# Number of occurrences of a given angle formed using 3 vertices of a n-sided regular polygon

Given an n-sided regular polygon and an angle θ, the task is to find number of occurrences of angle ( Ai, Aj, Ak ) = θ ( i < j < k) in a regular n-gon (regular polygon with n vertices) with vertices marked as A1, A2, …, An.

Examples:

```Input: n = 4, ang = 90
Output: 4

Input: n = 6, ang = 50
Output: 0
```

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

Approach:

1. First we check whether such an angle can exist or not.
2. Consider the vertices to be x, y, and z and the angle to find be xyz.
3. The number of edges between x and y be a and the number of edges between y and z be b.
4. Then xyz = 180 – (180*(a+b)) / n.
5. Thus xyz * n (mod 180) = 0.
6. Next we need to find the count of such angles.
7. As the polygon is regular we just need to calculate the count of such an angle at one vertex and can directly multiply our result by n (the number of vertices).
8. At each vertex the angle can be found at n-1-freq times where freq = (n*ang)/180 and depicts the number of edges remaining after creating the required angle i.e. the number of edges between z and x.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function that calculates occurrences ` `// of given angle that can be created ` `// using any 3 sides ` `int` `solve(``int` `ang, ``int` `n) ` `{ ` ` `  `    ``// Maximum angle in a regular n-gon ` `    ``// is equal to the interior angle ` `    ``// If the given angle ` `    ``// is greater than the interior angle ` `    ``// then the given angle cannot be created ` `    ``if` `((ang * n) > (180 * (n - 2))) { ` `        ``return` `0; ` `    ``} ` ` `  `    ``// The given angle times n should be divisible ` `    ``// by 180 else it cannot be created ` `    ``else` `if` `((ang * n) % 180 != 0) { ` `        ``return` `0; ` `    ``} ` ` `  `    ``// Initialise answer ` `    ``int` `ans = 1; ` ` `  `    ``// Calculate the frequency ` `    ``// of given angle for each vertex ` `    ``int` `freq = (ang * n) / 180; ` ` `  `    ``// Multiply answer by frequency. ` `    ``ans = ans * (n - 1 - freq); ` ` `  `    ``// Multiply answer by the number of vertices. ` `    ``ans = ans * n; ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `ang = 90, n = 4; ` ` `  `    ``cout << solve(ang, n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach ` `class` `GFG ` `{ ` ` `  `// Function that calculates occurrences ` `// of given angle that can be created ` `// using any 3 sides ` `static` `int` `solve(``int` `ang, ``int` `n) ` `{ ` ` `  `    ``// Maximum angle in a regular n-gon ` `    ``// is equal to the interior angle ` `    ``// If the given angle ` `    ``// is greater than the interior angle ` `    ``// then the given angle cannot be created ` `    ``if` `((ang * n) > (``180` `* (n - ``2``))) ` `    ``{ ` `        ``return` `0``; ` `    ``} ` ` `  `    ``// The given angle times n should be divisible ` `    ``// by 180 else it cannot be created ` `    ``else` `if` `((ang * n) % ``180` `!= ``0``) ` `    ``{ ` `        ``return` `0``; ` `    ``} ` ` `  `    ``// Initialise answer ` `    ``int` `ans = ``1``; ` ` `  `    ``// Calculate the frequency ` `    ``// of given angle for each vertex ` `    ``int` `freq = (ang * n) / ``180``; ` ` `  `    ``// Multiply answer by frequency. ` `    ``ans = ans * (n - ``1` `- freq); ` ` `  `    ``// Multiply answer by the number of vertices. ` `    ``ans = ans * n; ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main (String[] args) ` `{ ` `    ``int` `ang = ``90``, n = ``4``; ` `    ``System.out.println(solve(ang, n)); ` `} ` `} ` ` `  `// This code is contributed by Rajput-Ji `

## Python3

 `# Python3 implementation of the approach ` ` `  `# Function that calculates occurrences ` `# of given angle that can be created ` `# using any 3 sides ` `def` `solve(ang, n): ` ` `  `    ``# Maximum angle in a regular n-gon ` `    ``# is equal to the interior angle ` `    ``# If the given angle ` `    ``# is greater than the interior angle ` `    ``# then the given angle cannot be created ` `    ``if` `((ang ``*` `n) > (``180` `*` `(n ``-` `2``))): ` `        ``return` `0` ` `  `    ``# The given angle times n should be divisible ` `    ``# by 180 else it cannot be created ` `    ``elif` `((ang ``*` `n) ``%` `180` `!``=` `0``): ` `        ``return` `0` ` `  `    ``# Initialise answer ` `    ``ans ``=` `1` ` `  `    ``# Calculate the frequency ` `    ``# of given angle for each vertex ` `    ``freq ``=` `(ang ``*` `n) ``/``/` `180` ` `  `    ``# Multiply answer by frequency. ` `    ``ans ``=` `ans ``*` `(n ``-` `1` `-` `freq) ` ` `  `    ``# Multiply answer by the number of vertices. ` `    ``ans ``=` `ans ``*` `n ` ` `  `    ``return` `ans ` ` `  `# Driver code ` `ang ``=` `90` `n ``=` `4` ` `  `print``(solve(ang, n)) ` ` `  `# This code is contributed by Mohit Kumar `

## C#

 `// C# implementation of the approach ` `using` `System; ` `     `  `class` `GFG ` `{ ` ` `  `// Function that calculates occurrences ` `// of given angle that can be created ` `// using any 3 sides ` `static` `int` `solve(``int` `ang, ``int` `n) ` `{ ` ` `  `    ``// Maximum angle in a regular n-gon ` `    ``// is equal to the interior angle ` `    ``// If the given angle ` `    ``// is greater than the interior angle ` `    ``// then the given angle cannot be created ` `    ``if` `((ang * n) > (180 * (n - 2))) ` `    ``{ ` `        ``return` `0; ` `    ``} ` ` `  `    ``// The given angle times n should be divisible ` `    ``// by 180 else it cannot be created ` `    ``else` `if` `((ang * n) % 180 != 0) ` `    ``{ ` `        ``return` `0; ` `    ``} ` ` `  `    ``// Initialise answer ` `    ``int` `ans = 1; ` ` `  `    ``// Calculate the frequency ` `    ``// of given angle for each vertex ` `    ``int` `freq = (ang * n) / 180; ` ` `  `    ``// Multiply answer by frequency. ` `    ``ans = ans * (n - 1 - freq); ` ` `  `    ``// Multiply answer by the ` `    ``// number of vertices. ` `    ``ans = ans * n; ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main (String[] args) ` `{ ` `    ``int` `ang = 90, n = 4; ` `    ``Console.WriteLine(solve(ang, n)); ` `} ` `} ` ` `  `// This code is contributed by Princi Singh `

Output:

```4
```

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.