Biggest Square that can be inscribed within an Equilateral triangle

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 ` `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

 ``

Javascript

 ``

Output:

`2.32`

Time Complexity: O(1)

Auxiliary Space: O(1)

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