# Biggest Reuleaux Triangle inscirbed within a square inscribed in a semicircle

Given here is a semicircle of radius **r** 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 : x = 5Output : 14.0954Input : x = 8Output : 36.0842

**Approach**:We know, the side of the square inscribed within a semicircle is, **a = 2r/√5**. (Please refer here)

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

So, **x = 2*r/√5**

So, Area of Reuleaux Triangle:

A = 0.70477*x^2 = 0.70477*(r^2/5)

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 semicircle` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to find the biggest reuleaux triangle` `float` `Area(` `float` `r)` `{` ` ` `// radius cannot be negative` ` ` `if` `(r < 0)` ` ` `return` `-1;` ` ` `// height of the reuleaux triangle` ` ` `float` `x = (2 * r) / ` `sqrt` `(5);` ` ` `// area of the reuleaux triangle` ` ` `float` `A = 0.70477 * ` `pow` `(x, 2);` ` ` `return` `A;` `}` `// Driver code` `int` `main()` `{` ` ` `float` `r = 5;` ` ` `cout << Area(r) << endl;` ` ` `return` `0;` `}` |

## Java

`// Java Program to find the biggest` `// Reuleaux triangle inscribed within` `// in a square which in turn is` `// inscribed within a semicircle` `import` `java.lang.Math;` `class` `GFG` `{` `// Function to find the biggest reuleaux triangle` `static` `float` `Area(` `float` `r)` `{` ` ` `// radius cannot be negative` ` ` `if` `(r < ` `0` `)` ` ` `return` `-` `1` `;` ` ` `// height of the reuleaux triangle` ` ` `float` `x = (` `2` `* r) /(` `float` `)(Math.sqrt(` `5` `));` ` ` `// 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` `r = ` `5` `;` ` ` `System.out.println(Area(r));` `}` `}` `// This code is contributed by Mukul Singh.` |

## Python3

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

## C#

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

## PHP

`<?php` `// PHP Program to find the biggest Reuleaux` `// triangle inscribed within in a square` `// which in turn is inscribed within a semicircle` `// Function to find the biggest` `// reuleaux triangle` `function` `Area(` `$r` `)` `{` ` ` `// radius cannot be negative` ` ` `if` `(` `$r` `< 0)` ` ` `return` `-1;` ` ` `// height of the reuleaux triangle` ` ` `$x` `= (2 * ` `$r` `) / sqrt(5);` ` ` `// area of the reuleaux triangle` ` ` `$A` `= 0.70477 * pow(` `$x` `, 2);` ` ` `return` `$A` `;` `}` `// Driver code` `$r` `= 5;` `echo` `Area(` `$r` `);` ` ` `// This code is contributed by Ryuga` `?>` |

## Javascript

`<script>` `// javascript Program to find the biggest` `// Reuleaux triangle inscribed within` `// in a square which in turn is` `// inscribed within a semicircle` `// Function to find the biggest reuleaux triangle` `function` `Area(r)` `{` ` ` `// radius cannot be negative` ` ` `if` `(r < 0)` ` ` `return` `-1;` ` ` `// height of the reuleaux triangle` ` ` `var` `x = (2 * r) /(Math.sqrt(5));` ` ` `// area of the reuleaux triangle` ` ` `var` `A = 0.70477 *(Math.pow(x, 2));` ` ` `return` `A;` `}` `// Driver code` `var` `r = 5;` `document.write(Area(r).toFixed(4));` `// This code is contributed by Princi Singh` `</script>` |

**Output:**

14.0954

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