# 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; ` `} ` |

*chevron_right*

*filter_none*

## 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 ` |

*chevron_right*

*filter_none*

## 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 ` |

*chevron_right*

*filter_none*

## 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
[tabby title="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 ` |

*chevron_right*

*filter_none*

**Output:**

3.79335

## Recommended Posts:

- Biggest Reuleaux Triangle inscribed within a square which is inscribed within a hexagon
- Biggest Reuleaux Triangle inscribed within a square which is inscribed within an ellipse
- Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle
- Biggest Square that can be inscribed within an Equilateral triangle
- Biggest Reuleaux Triangle within a Square which is inscribed within a Circle
- Biggest Reuleaux Triangle inscirbed within a square inscribed in a semicircle
- Largest square that can be inscribed within a hexagon which is inscribed within an equilateral triangle
- Area of a square inscribed in a circle which is inscribed in an equilateral triangle
- Biggest Reuleaux Triangle within A Square
- Largest hexagon that can be inscribed within an equilateral triangle
- Area of circle which is inscribed in equilateral triangle
- Count of distinct rectangles inscribed in an equilateral triangle
- Area of a triangle inscribed in a rectangle which is inscribed in an ellipse
- Radius of the biggest possible circle inscribed in rhombus which in turn is inscribed in a rectangle
- Largest triangle that can be inscribed in an ellipse

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.