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Biggest Square that can be inscribed within an Equilateral triangle
  • Last Updated : 14 Oct, 2019

Given here is an equilateral triangle of side length a. The task is to find the side of the biggest square that can be inscribed within it.

Examples:

Input: a = 5 
Output: 2.32

Input: a = 7
Output: 3.248

Approach: Let the side of the square be x.
Now, AH is perpendicular to DE.
DE is parallel to BC, So, angle AED = angle ACB = 60

In triangle EFC,
              => Sin60 = x/ EC
              => √3 / 2 = x/EC
              => EC = 2x/√3
In triangle AHE,
              => Cos 60 = x/2AE
              => 1/2 = x/2AE
              => AE = x

So, side AC of the triangle = 2x/√3 + x. Now,
a = 2x/√3 + x
Therefore, x = a/(1 + 2/√3) = 0.464a



Below is the implementation of the above approach:

C++




// C++ Program to find the biggest square
// which can be inscribed within the equilateral triangle
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the side
// of the square
float square(float a)
{
  
    // the side cannot be negative
    if (a < 0)
        return -1;
  
    // side of the square
    float x = 0.464 * a;
  
    return x;
}
  
// Driver code
int main()
{
    float a = 5;
    cout << square(a) << endl;
  
    return 0;
}


Java




// Java Program to find the 
// the biggest square which
// can be inscribed within 
// the equilateral triangle
  
class GFG
{
    // Function to find the side
    // of the square
    static double square(double a)
    {
      
        // the side cannot be negative
        if (a < 0)
            return -1;
      
        // side of the square
        double x = 0.464 * a;
        return x;
    }
      
    // Driver code
    public static void main(String []args)
    {
        double a = 5;
        System.out.println(square(a));
    }
}
  
// This code is contributed by ihritik


Python3




# Python3 Program to find the biggest square
# which can be inscribed within the equilateral triangle
  
# Function to find the side
# of the square
def square( a ):
  
  
    # the side cannot be negative
    if (a < 0):
        return -1
  
    # side of the square
    x = 0.464 * a
  
    return x
  
  
# Driver code
a = 5
print(square(a))
  
# This code is contributed by ihritik


C#




// C# Program to find the biggest 
// square which can be inscribed 
// within the equilateral triangle
using System;
  
class GFG
{
    // Function to find the side
    // of the square
    static double square(double a)
    {
      
        // the side cannot be negative
        if (a < 0)
            return -1;
      
        // side of the square
        double x = 0.464 * a;
        return x;
    }
      
    // Driver code
    public static void Main()
    {
        double a = 5;
        Console.WriteLine(square(a));
    }
}
  
// This code is contributed by ihritik


PHP




<?php
// PHP Program to find the biggest
//  square which can be inscribed 
// within the equilateral triangle
  
// Function to find the side
// of the square
function square($a )
{
  
    // the side cannot be negative
    if ($a < 0)
        return -1;
  
    // side of the square
    $x = 0.464 * $a;
    return $x;
}
  
// Driver code
$a = 5;
echo square($a);
  
// This code is contributed by ihritik
  
?>


Output:

2.32

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