# Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle

Given here is a right angle triangle with height **l**, base **b** & hypotenuse **h**, 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:l = 5, b = 12, h = 13Output:8.77914Input:l = 3, b = 4, h = 5Output:2.07116

**Approach**: We know, the side of the square inscribed within a right angled triangle is, **a = (l*b)/(l+b)**, please refer Area of a largest square fit in a right angle triangle.

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

So, **x = (l*b)/(l+b)**.

So, Area of the Reuleaux Triangle is**, A = 0.70477*x^2 = 0.70477*((l*b)/(l+b))^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 circle ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find the biggest reuleaux triangle ` `float` `Area(` `float` `l, ` `float` `b, ` `float` `h) ` `{ ` ` ` ` ` `// the height or base or hypotenuse ` ` ` `// cannot be negative ` ` ` `if` `(l < 0 || b < 0 || h < 0) ` ` ` `return` `-1; ` ` ` ` ` `// height of the reuleaux triangle ` ` ` `float` `x = (l * b) / (l + b); ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `float` `A = 0.70477 * ` `pow` `(x, 2); ` ` ` ` ` `return` `A; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `float` `l = 5, b = 12, h = 13; ` ` ` `cout << Area(l, b, h) << 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 circle ` `import` `java.util.*; ` `import` `java.text.DecimalFormat; ` ` ` `class` `GFG ` `{ ` ` ` `// Function to find the biggest reuleaux triangle ` `static` `double` `Area(` `double` `l, ` `double` `b, ` `double` `h) ` `{ ` ` ` ` ` `// the height or base or hypotenuse ` ` ` `// cannot be negative ` ` ` `if` `(l < ` `0` `|| b < ` `0` `|| h < ` `0` `) ` ` ` `return` `-` `1` `; ` ` ` ` ` `// height of the reuleaux triangle ` ` ` `double` `x = (l * b) / (l + b); ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `double` `A = ` `0.70477` `* Math.pow(x, ` `2` `); ` ` ` ` ` `return` `A; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String args[]) ` `{ ` ` ` `double` `l = ` `5` `, b = ` `12` `, h = ` `13` `; ` ` ` `DecimalFormat df = ` `new` `DecimalFormat(` `"#,###,##0.00000"` `); ` ` ` `System.out.println(df.format(Area(l, b, h))); ` `} ` `} ` ` ` `// This code is contributed by ` `// Shashank_Sharma ` |

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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 circle ` `import` `math as mt ` ` ` `# Function to find the biggest ` `# reuleaux triangle ` `def` `Area(l, b, h): ` ` ` ` ` `# the height or base or hypotenuse ` ` ` `# cannot be negative ` ` ` `if` `(l < ` `0` `or` `b < ` `0` `or` `h < ` `0` `): ` ` ` `return` `-` `1` ` ` ` ` `# height of the reuleaux triangle ` ` ` `x ` `=` `(l ` `*` `b) ` `/` `(l ` `+` `b) ` ` ` ` ` `# area of the reuleaux triangle ` ` ` `A ` `=` `0.70477` `*` `pow` `(x, ` `2` `) ` ` ` ` ` `return` `A ` ` ` `# Driver code ` `l, b, h ` `=` `5` `, ` `12` `, ` `13` `print` `(Area(l, b, h)) ` ` ` `# 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 a circle ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Function to find the biggest reuleaux triangle ` `static` `double` `Area(` `double` `l, ` `double` `b, ` `double` `h) ` `{ ` ` ` ` ` `// the height or base or hypotenuse ` ` ` `// cannot be negative ` ` ` `if` `(l < 0 || b < 0 || h < 0) ` ` ` `return` `-1; ` ` ` ` ` `// height of the reuleaux triangle ` ` ` `double` `x = (l * b) / (l + b); ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `double` `A = 0.70477 * Math.Pow(x, 2); ` ` ` ` ` `return` `A; ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main() ` `{ ` ` ` `double` `l = 5, b = 12, h = 13; ` ` ` `Console.WriteLine((Area(l, b, h))); ` `} ` `} ` ` ` `// This code is contributed by ` `// Mukul Singh ` |

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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 circle ` ` ` `// Function to find the biggest ` `// reuleaux triangle ` `function` `Area(` `$l` `, ` `$b` `, ` `$h` `) ` `{ ` ` ` ` ` `// the height or base or hypotenuse ` ` ` `// cannot be negative ` ` ` `if` `(` `$l` `< 0 ` `or` `$b` `< 0 ` `or` `$h` `< 0) ` ` ` `return` `-1; ` ` ` ` ` `// height of the reuleaux triangle ` ` ` `$x` `= (` `$l` `* ` `$b` `) / (` `$l` `+ ` `$b` `); ` ` ` ` ` `// area of the reuleaux triangle ` ` ` `$A` `= 0.70477 * pow(` `$x` `, 2); ` ` ` ` ` `return` `$A` `; ` `} ` ` ` `// Driver code ` `$l` `= 5; ` `$b` `= 12; ` `$h` `= 13; ` `echo` `Area(` `$l` `, ` `$b` `, ` `$h` `); ` ` ` `// This code is contributed by ` `// anuj_67 ` `?> ` |

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**Output:**

8.77914

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