# Biggest Reuleaux Triangle inscribed within a square which is inscribed within a hexagon

Given a regular hexagon of side length a which inscribes a square which in turn inscribes a reuleaux triangle. The task is to find the maximum possible area of this reuleaux triangle.

Examples:

```Input: a = 5
Output: 28.3287

Input: a = 9
Output: 91.7848
``` ## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: As the side of the square inscribed within a hexagon is x = 1.268a. Please refer Largest Square that can be inscribed within a hexagon.
Also, in the reuleaux triangle, h = x = 1.268a.
So, Area of the reuleaux triangle, A = 0.70477*h^2 = 0.70477*(1.268a)^2.

Below is the implementation of the above approach:

## C++

 `// C++ Program to find the biggest Reuleaux triangle ` `// inscribed within in a square which in turn ` `// is inscribed within a hexagon ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the biggest reuleaux triangle ` `float` `Area(``float` `a) ` `{ ` ` `  `    ``// side cannot be negative ` `    ``if` `(a < 0) ` `        ``return` `-1; ` ` `  `    ``// height of the reuleaux triangle ` `    ``float` `h = 1.268 * a; ` ` `  `    ``// area of the reuleaux triangle ` `    ``float` `A = 0.70477 * ``pow``(h, 2); ` ` `  `    ``return` `A; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``float` `a = 5; ` `    ``cout << Area(a) << endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java Program to find the biggest Reuleaux triangle ` `// inscribed within in a square which in turn ` `// is inscribed within a hexagon ` ` `  `import` `java.io.*; ` ` `  `class` `GFG  ` `{ ` ` `  `// Function to find the biggest reuleaux triangle ` `static` `float` `Area(``float` `a) ` `{ ` ` `  `    ``// side cannot be negative ` `    ``if` `(a < ``0``) ` `        ``return` `-``1``; ` ` `  `    ``// height of the reuleaux triangle ` `    ``float` `h =(``float``) ``1.268` `* a; ` ` `  `    ``// area of the reuleaux triangle ` `    ``float` `A = (``float``)(``0.70477` `* Math.pow(h, ``2``)); ` ` `  `    ``return` `A; ` `} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `        ``float` `a = ``5``; ` `    ``System.out.println( Area(a)); ` `    ``} ` `} ` ` `  `// This code is contributed by anuj_67 `

## Python3

 `# Python3 Program to find the biggest  ` `# Reuleaux triangle inscribed within ` `# in a square which in turn is ` `# inscribed within a hexagon ` `import` `math ` ` `  `# Function to find the biggest  ` `# reuleaux triangle ` `def` `Area(a): ` ` `  `    ``# side cannot be negative ` `    ``if` `(a < ``0``): ` `        ``return` `-``1` ` `  `    ``# height of the reuleaux triangle ` `    ``h ``=` `1.268` `*` `a ` ` `  `    ``# area of the reuleaux triangle ` `    ``A ``=` `0.70477` `*` `math.``pow``(h, ``2``) ` ` `  `    ``return` `A ` ` `  `# Driver code ` `a ``=` `5` `print``(Area(a),end ``=` `"\n"``) ` ` `  `# This code is contributed  ` `# by Akanksha Rai `

## C#

 `// C# Program to find the biggest Reuleaux ` `// triangle inscribed within in a square  ` `// which in turn is inscribed within a hexagon ` `using` `System; ` ` `  `class` `GFG  ` `{ ` ` `  `// Function to find the biggest reuleaux triangle ` `static` `float` `Area(``float` `a) ` `{ ` ` `  `    ``// side cannot be negative ` `    ``if` `(a < 0) ` `        ``return` `-1; ` ` `  `    ``// height of the reuleaux triangle ` `    ``float` `h =(``float``) 1.268 * a; ` ` `  `    ``// area of the reuleaux triangle ` `    ``float` `A = (``float``)(0.70477 * Math.Pow(h, 2)); ` ` `  `    ``return` `A; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main ()  ` `{ ` `    ``float` `a = 5; ` `    ``Console.WriteLine(Area(a)); ` `} ` `} ` ` `  `// This code is contributed  ` `// by Akanksha Rai `

## PHP

 ` `

Output:

```28.3287
```

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Improved By : vt_m, AnkitRai01, Akanksha_Rai