# Biggest Reuleaux Triangle inscribed within a Square inscribed in an equilateral triangle

Given here is an equilateral triangle of sidelength **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 : 3.79335Input : a = 9Output : 12.2905

**Approach**: We know that the side of the square inscribed within an equilateral triangle of side length is, **x = 0.464*a** (Please refer here).

Also, in the reuleaux triangle, **h = x**.

So, Area of Reuleaux Triangle:

A= 0.70477*h^{2}= 0.70477*(0.464*a)^{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 an equilateral triangle ` `#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` `x = 0.464 * a; ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `float` `A = 0.70477 * ` `pow` `(x, 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 an equilateral triangle ` ` ` `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` `x = ` `0` `.464f * a; ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `float` `A = ` `0` `.70477f * (` `float` `)Math.pow(x, ` `2` `); ` ` ` ` ` `return` `A; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main (String[] args) ` `{ ` ` ` `float` `a = ` `5` `; ` ` ` `System.out.println(String.format(` `"%.5f"` `, Area(a))); ` `} ` `} ` ` ` `// This code is contributed by chandan_jnu ` |

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## Python3

`# Python3 Program to find the biggest ` `# Reuleaux triangle inscribed within ` `# in a square which in turn is inscribed ` `# within an equilateral triangle ` `import` `math as mt ` ` ` `# Function to find the biggest ` `# reuleaux triangle ` `def` `Area(a): ` ` ` ` ` `# side cannot be negative ` ` ` `if` `(a < ` `0` `): ` ` ` `return` `-` `1` ` ` ` ` `# height of the reuleaux triangle ` ` ` `x ` `=` `0.464` `*` `a ` ` ` ` ` `# area of the reuleaux triangle ` ` ` `A ` `=` `0.70477` `*` `pow` `(x, ` `2` `) ` ` ` ` ` `return` `A ` ` ` `# Driver code ` `a ` `=` `5` `print` `(Area(a)) ` ` ` `# This code is contributed by ` `# Mohit Kumar 29 ` |

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## C#

`// C# Program to find the biggest Reuleaux ` `// triangle inscribed within in a square ` `// which in turn is inscribed within an ` `// equilateral triangle ` `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` `x = 0.464f * a; ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `float` `A = 0.70477f * (` `float` `)Math.Pow(x, 2); ` ` ` ` ` `return` `A; ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main () ` `{ ` ` ` `float` `a = 5; ` ` ` `Console.WriteLine(String.Format(` `"{0,0:#.00000}"` `, ` ` ` `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 an ` `// equilateral triangle ` ` ` `// Function to find the biggest ` `// reuleaux triangle ` `function` `Area(` `$a` `) ` `{ ` ` ` ` ` `// side cannot be negative ` ` ` `if` `(` `$a` `< 0) ` ` ` `return` `-1; ` ` ` ` ` `// height of the reuleaux triangle ` ` ` `$x` `= 0.464 * ` `$a` `; ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `$A` `= 0.70477 * pow(` `$x` `, 2); ` ` ` ` ` `return` `$A` `; ` `} ` ` ` `// Driver code ` `$a` `= 5; ` `echo` `Area(` `$a` `) . ` `"\n"` `; ` ` ` `// This code is contributed ` `// by Akanksha Rai ` |

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**Output:**

3.79335

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