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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 Square.
So, Area of the Reuleaux Triangle is, A = 0.70477*h^2 = 0.70477*2*r^2

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 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; 
} 

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.*; 
  
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 

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 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 

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 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.. 

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 Area of the  
// Reuleaux triangle 
function 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 
$r = 6; 
echo ReuleauxArea($r) . "\n"; 
  
// This code is contributed by ita_c 
?> 

Output:

50.7434

My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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