Largest hexagon that can be inscribed within a square

Given side of a square a, the task is to find the side of the largest hexagon that can be inscribed within the given square.
Examples:

Input: a = 6 
Output: 3.1056
Input: a = 8 
Output: 4.1408 

Approach:: Let, the side of the hexagon be x and assume that the side of the square, a gets divided into smaller length b & bigger length c i.e. a = b + c
Now from the figure, we see,

b² + b² = x² which gives b = x / ?2
Now again, d / (2 * x) = cos(30) = ?3 / 2 i.e. d = x?3
And, c² + c² = d² which gives c = d / ?2 = x?3 / ?2
Since, a = b + c. So, a = x / ?2 + x?3 / ?2 = ((1 + ?3) / ?2) * x = 1.932 * x
So, side of the hexagon, x = 0.5176 * a

Below is the implementation of the above approach:

C++

// C++ Program to find the biggest hexagon which
// can be inscribed within the given square
#include <bits/stdc++.h>
using namespace std;

// Function to return the side
// of the hexagon
float hexagonside(float a)
{

    // Side cannot be negative
    if (a < 0)
        return -1;

    // Side of the hexagon
    float x = 0.5176 * a;
    return x;
}

// Driver code
int main()
{
    float a = 6;
    cout << hexagonside(a) << endl;
    return 0;
}

Java

// Java  Program to find the biggest hexagon which
// can be inscribed within the given square

import java.io.*;

class GFG {

// Function to return the side
// of the hexagon
static double hexagonside(double a)
{

    // Side cannot be negative
    if (a < 0)
        return -1;

    // Side of the hexagon
    double x = (0.5176 * a);
    return x;
}

// Driver code
    public static void main (String[] args) {

        double a = 6;
        System.out.println (hexagonside(a));
    }
//This code is contributed by ajit.
}

Python 3

# Python 3 Program to find the biggest
# hexagon which can be inscribed within
# the given square

# Function to return the side
# of the hexagon
def hexagonside(a):

    # Side cannot be negative
    if (a < 0):
        return -1;

    # Side of the hexagon
    x = 0.5176 * a;
    return x;

# Driver code
a = 6;
print(hexagonside(a));

# This code is contributed
# by Akanksha Rai

C#

// C# Program to find the biggest hexagon which
// can be inscribed within the given square
using System;

class GFG
{

// Function to return the side
// of the hexagon
static double hexagonside(double a)
{

    // Side cannot be negative
    if (a < 0)
        return -1;

    // Side of the hexagon
    double x = (0.5176 * a);
    return x;
}

// Driver code
public static void Main ()
{
    double a = 6;
    Console.WriteLine(hexagonside(a));
}
}

// This code is contributed by Ryuga.

PHP

<?php
// PHP Program to find the biggest hexagon which
// can be inscribed within the given square

// Function to return the side of the hexagon
function hexagonside($a)
{

    // Side cannot be negative
    if ($a < 0)
        return -1;

    // Side of the hexagon
    $x = 0.5176 * $a;
    return $x;
}

// Driver code
$a = 6;
echo hexagonside($a);

// This code is contributed by akt_mit
?>

Javascript

<script>

// Javascript Program to find the biggest hexagon which
// can be inscribed within the given square

// Function to return the side
// of the hexagon
function hexagonside(a)
{

    // Side cannot be negative
    if (a < 0)
        return -1;

    // Side of the hexagon
    let x = 0.5176 * a;
    return x;
}

// Driver code

    let a = 6;
    document.write(hexagonside(a) + "<br>");

// This code is contributed by Manoj

</script>

Output:

3.1056

Time Complexity: O(1)

Auxiliary Space: O(1)