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

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  • Last Updated : 28 Jul, 2022

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 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
?>

Javascript




<script>
// javascript 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
    var A = 0.70477 * 2 * Math.pow(r, 2);
 
    return A;
}
 
// Driver code
 
var r = 6;
document.write(ReuleauxArea(r));
 
 
// This code contributed by Princi Singh
</script>

Output: 

50.7434

 

Time Complexity: O(1)

Auxiliary Space: O(1)


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