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Number of occurrences of a given angle formed using 3 vertices of a n-sided regular polygon
• Last Updated : 07 Apr, 2021

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```

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`

## Javascript

 ``
Output:
`4`

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