Largest Square that can be inscribed within a hexagon

Given a regular hexagonof side length a, the task is to find the area of the largest square that can be inscribed within it.

Examples:

Input: a = 6
Output: 57.8817

Input: a = 8
Output: 102.901



Approach: The square we will derive will have the same centre and axes of the hexagon. This is because the square will become smaller if we will rotate it.

The sides of the hexagonal are equal i.e. a = b + c.
Now, let d be the length of the side of the inscribed square,
Then the top side of the square, d = 2 * c * sin(60).
And, the left side of the square, d = a + 2 * b * sin(30).
Substituting for c, d = 2 * (a – b) * sin(60).
Now taking d and re-arranging, we get, b / a = (2 * sin(60) – 1) / (2 * (sin(30) + sin(60)))
So, b / a = 2 – √3
Now, substituting the relation of b and a in the left hand side equation of square, we get,
d / a = 3 – √3 i.e. d / a = 1.268
Therefore, d = 1.268 * a

Below is the implementation of the above approach:

C++

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// C++ program to find the area of the largest square
// that can be inscribed within the hexagon
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the area
// of the square
float squareArea(float a)
{
  
    // Side cannot be negative
    if (a < 0)
        return -1;
  
    // Area of the square
    float area = pow(1.268, 2) * pow(a, 2);
    return area;
}
  
// Driver code
int main()
{
    float a = 6;
    cout << squareArea(a) << endl;
    return 0;
}

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Java

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// Java program to find the area of the largest square
// that can be inscribed within the hexagon
class Solution {
    // Function to find the area
    // of the square
    static float squareArea(float a)
    {
  
        // Side cannot be negative
        if (a < 0)
            return -1;
  
        // Area of the square
        float area = (float)(Math.pow(1.268, 2) * Math.pow(a, 2));
        return area;
    }
  
    // Driver code
    public static void main(String args[])
    {
        float a = 6;
        System.out.println(squareArea(a));
    }
}
  
// This code is contributed by Arnab Kundu

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Python3

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# Python program to find the area of the largest square
# that can be inscribed within the hexagon
  
# Function to find the area
# of the square
def squareArea(a):
  
    # Side cannot be negative
    if (a < 0):
        return -1;
  
    # Area of the square
    area = (1.268 ** 2) * (a ** 2);
    return area;
  
# Driver code
a = 6;
print(squareArea(a));
  
# This code contributed by PrinciRaj1992 

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C#

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// C# program to find the area of the largest square
// that can be inscribed within the hexagon
using System;
class Solution {
    // Function to find the area
    // of the square
    static float squareArea(float a)
    {
  
        // Side cannot be negative
        if (a < 0)
            return -1;
  
        // Area of the square
        float area = (float)(Math.Pow(1.268, 2) * Math.Pow(a, 2));
        return area;
    }
  
    // Driver code
    public static void Main()
    {
        float a = 6;
        Console.WriteLine(squareArea(a));
    }
}
  
// This code is contributed by  anuj_67..

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PHP

Output:

57.8817


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