# 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 = 5Output:28.3287Input:a = 9Output:91.7848

**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 <bits/stdc++.h> ` `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; ` `} ` |

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## 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 ` |

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## 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
[tabby title="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 ` |

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## PHP

`<?php ` `// PHP Program to find the biggest Reuleaux ` `// triangle inscribed within in a square ` `// which in turn is inscribed within a hexagon ` ` ` `// Function to find the biggest ` `// reuleaux triangle ` `function` `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 ` ` ` `$A` `= 0.70477 * pow(` `$h` `, 2); ` ` ` ` ` `return` `$A` `; ` `} ` ` ` `// Driver code ` `$a` `= 5; ` `echo` `round` `(Area(` `$a` `), 4); ` ` ` `// This code is contributed by Ryuga ` `?> ` |

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**Output:**

28.3287

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