Biggest Reuleaux Triangle inscribed within a square which is inscribed within an ellipse

Given an ellipse with major axis length and minor axis 2a & 2b respectively 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 = 5, b = 4
Output: 0.0722389

Input: a = 7, b = 11
Output: 0.0202076

Approach: As, the side of the square inscribed within an ellipse is, x = √(a^2 + b^2)/ab. Please refer Area of the Largest square that can be inscribed in an ellipse.
Also, in the reuleaux triangle, h = x = √(a^2 + b^2)/ab. 
So, Area of the reuleaux triangle, A = 0.70477*h^2 = 0.70477*((a^2 + b^2)/a^2b^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 ellipse
#include <bits/stdc++.h>
using namespace std;

// Function to find the biggest reuleaux triangle
float Area(float a, float b)
{

    // length of the axes cannot be negative
    if (a < 0 && b < 0)
        return -1;

    // height of the reuleaux triangle
    float h = sqrt(((pow(a, 2) + pow(b, 2))
                    / (pow(a, 2) * pow(b, 2))));

    // area of the reuleaux triangle
    float A = 0.70477 * pow(h, 2);

    return A;
}

// Driver code
int main()
{
    float a = 5, b = 4;
    cout << Area(a, b) << endl;

    return 0;
}

Java

// Java Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within an ellipse
import java.io.*;

class GFG
{

// Function to find the biggest reuleaux triangle
static float Area(float a, float b)
{

    // length of the axes cannot be negative
    if (a < 0 && b < 0)
        return -1;

    // height of the reuleaux triangle
    float h = (float)Math.sqrt(((Math.pow(a, 2) + Math.pow(b, 2))
                / (Math.pow(a, 2) * Math.pow(b, 2))));

    // 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, b = 4;
    System.out.println(Area(a, b));
}
}

// This code is contributed by anuj_67..

Python3

# Python3 Program to find the biggest Reuleaux
# triangle inscribed within in a square
# which in turn is inscribed within an ellipse
import math;

# Function to find the biggest
# reuleaux triangle
def Area(a, b):

    # length of the axes cannot
    # be negative
    if (a < 0 and b < 0):
        return -1;

    # height of the reuleaux triangle
    h = math.sqrt(((pow(a, 2) + pow(b, 2)) /
                   (pow(a, 2) * pow(b, 2))));

    # area of the reuleaux triangle
    A = 0.70477 * pow(h, 2);

    return A;

# Driver code
a = 5;
b = 4;
print(round(Area(a, b), 7));

# This code is contributed by chandan_jnu

C#

// C# Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within an ellipse
using System;

class GFG
{

// Function to find the biggest reuleaux triangle
static double Area(double a, double b)
{

    // length of the axes cannot be negative
    if (a < 0 && b < 0)
        return -1;

    // height of the reuleaux triangle
    double h = (double)Math.Sqrt(((Math.Pow(a, 2) +
                                    Math.Pow(b, 2)) /
                                   (Math.Pow(a, 2) *
                                   Math.Pow(b, 2))));

    // area of the reuleaux triangle
    double A = (double)(0.70477 * Math.Pow(h, 2));

    return A;
}

// Driver code
static void Main()
{
    double a = 5, b = 4;
    Console.WriteLine(Math.Round(Area(a, b),7));
}
}

// 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 an ellipse

// Function to find the biggest
// reuleaux triangle
function Area($a, $b)
{

    // length of the axes cannot
    // be negative
    if ($a < 0 && $b < 0)
        return -1;

    // height of the reuleaux triangle
    $h = sqrt(((pow($a, 2) + pow($b, 2)) /
               (pow($a, 2) * pow($b, 2))));

    // area of the reuleaux triangle
    $A = 0.70477 * pow($h, 2);

    return $A;
}

// Driver code
$a = 5;
$b = 4;
echo round(Area($a, $b), 7);

// 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 an ellipse

// Function to find the biggest reuleaux triangle
function Area(a, b)
{

    // length of the axes cannot be negative
    if (a < 0 && b < 0)
        return -1;

    // height of the reuleaux triangle
    let h = Math.sqrt(((Math.pow(a, 2) + Math.pow(b, 2))
                    / (Math.pow(a, 2) * Math.pow(b, 2))));

    // area of the reuleaux triangle
    let A = 0.70477 * Math.pow(h, 2);

    return A;
}

// Driver code
    let a = 5, b = 4;
    document.write(Area(a, b) + "<br>");

// This code is contributed by Mayank Tyagi

</script>

Output:

0.0722389

Time Complexity: O(logn) as it is using inbuilt sqrt function 

Auxiliary Space: O(1)