Biggest Reuleaux Triangle within a Square which is inscribed within a Circle

Given here is a circle 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: r = 6
Output: 50.7434

Input: r = 11
Output: 170.554



Approach: From the figure, it is very clear that, if the side of the square is a, then

a√2 = 2r
a = √2r

Also, in reuleaux triangle, h = a = √2r, please refer Biggest Reuleaux Triangle within A Sqaure.
So, Area of the Reuleaux Triangle is, A = 0.70477*h^2 = 0.70477*2*r^2

C++

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// 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 Area
// of the Reuleaux triangle
float ReuleauxArea(float r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // Area of the Reuleaux triangle
    float A = 0.70477 * 2 * pow(r, 2);
  
    return A;
}
  
// Driver code
int main()
{
    float r = 6;
    cout << ReuleauxArea(r) << endl;
    return 0;
}

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Java

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// Java Program to find the biggest Reuleaux
// triangle inscribed within in a square which
// in turn is inscribed within a circle
import java.util.*;
  
class GFG
{
  
// Function to find the Area
// of the Reuleaux triangle
static double ReuleauxArea(double r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // Area of the Reuleaux triangle
    double A = 0.70477 * 2 * Math.pow(r, 2);
  
    return A;
}
  
// Driver code
public static void main(String args[])
{
    double r = 6;
    System.out.println(ReuleauxArea(r));
      
}
}
// This code is contributed by
// Surendra_Gangwar

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Python3

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# 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 Area
# of the Reuleaux triangle
def ReuleauxArea(r):
  
    # radius cannot be negative
    if (r < 0):
        return -1
  
    # Area of the Reuleaux triangle
    A = 0.70477 * 2 * pow(r, 2)
  
    return A
  
# Driver code
r = 6
print(ReuleauxArea(r))
  
# This code is contributed by 
# Mohit kumar 29

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C#

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// 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 Area
// of the Reuleaux triangle
static double ReuleauxArea(double r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // Area of the Reuleaux triangle
    double A = 0.70477 * 2 * Math.Pow(r, 2);
  
    return A;
}
  
// Driver code
public static void Main()
{
    double r = 6;
    Console.WriteLine(ReuleauxArea(r));
}
}
  
// This code is contributed by
// shs..

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PHP

Output:

50.7434


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