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

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++

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// C++ implementation of the approach
  
#include <bits/stdc++.h>
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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

4


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