Biggest Reuleaux Triangle inscirbed within a square inscribed in a semicircle

Given here is a semicircle 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 : x = 5  
Output : 14.0954

Input : x = 8
Output : 36.0842



Approach:We know, the side of the square inscribed within a semicircle is, a = 2r/√5. (Please refer here)

Also, in the reuleaux triangle, x = a.
So, x = 2*r/√5
So, Area of Reuleaux Triangle:

A = 0.70477*x^2 = 0.70477*(r^2/5)

Below is the implementation of the above approach:

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 semicircle
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the biggest reuleaux triangle
float Area(float r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // height of the reuleaux triangle
    float x = (2 * r) / sqrt(5);
  
    // area of the reuleaux triangle
    float A = 0.70477 * pow(x, 2);
  
    return A;
}
  
// Driver code
int main()
{
    float r = 5;
    cout << Area(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 semicircle
import java.lang.Math;
  
class GFG
{
  
// Function to find the biggest reuleaux triangle
static float Area(float r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // height of the reuleaux triangle
    float x = (2 * r) /(float)(Math.sqrt(5));
  
    // 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 r = 5;
    System.out.println(Area(r));
}
}
  
// This code is contributed by Mukul Singh.

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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 semicircle
import math as mt
  
# Function to find the biggest
# reuleaux triangle
def Area(r):
  
    # radius cannot be negative
    if (r < 0):
        return -1
  
    # height of the reuleaux triangle
    x = (2 * r) / mt.sqrt(5)
  
    # area of the reuleaux triangle
    A = 0.70477 * pow(x, 2)
  
    return A
  
# Driver code
r = 5
print(Area(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 semicircle
using System;
  
class GFG
{
  
// Function to find the biggest reuleaux triangle
static double Area(double r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // height of the reuleaux triangle
    double x = (2 * r) / (double)(Math.Sqrt(5));
  
    // area of the reuleaux triangle
    double A = 0.70477  * (double)(Math.Pow(x, 2));
  
    return A;
}
  
// Driver code
public static void Main()
{
    double r = 5;
    Console.WriteLine(Area(r));
}
}
  
// This code is contributed by chandan_jnu

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PHP

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<?php
// PHP Program to find the biggest Reuleaux 
// triangle inscribed within in a square 
// which in turn is inscribed within a semicircle
  
// Function to find the biggest 
// reuleaux triangle
function Area($r
{
  
    // radius cannot be negative
    if ($r < 0) 
        return -1;
  
    // height of the reuleaux triangle
    $x = (2 * $r) / sqrt(5);
  
    // area of the reuleaux triangle
    $A = 0.70477 * pow($x, 2);
  
    return $A ;
}
  
// Driver code
$r = 5;
  
echo Area($r);
      
// This code is contributed by Ryuga
?>
  

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Output:

14.0954


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