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Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle
  • Last Updated : 16 Mar, 2021

Given here is a right angle triangle with height l, base b & hypotenuse h, 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: l = 5, b = 12, h = 13
Output: 8.77914

Input: l = 3, b = 4, h = 5
Output: 2.07116

 

 

Approach: We know, the side of the square inscribed within a right angled triangle is, a = (l*b)/(l+b), please refer Area of a largest square fit in a right angle triangle
Also, in the reuleaux triangle, x = a
So, x = (l*b)/(l+b)
So, Area of the Reuleaux Triangle is, A = 0.70477*x^2 = 0.70477*((l*b)/(l+b))^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 a circle
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the biggest reuleaux triangle
float Area(float l, float b, float h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    float x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    float A = 0.70477 * pow(x, 2);
 
    return A;
}
 
// Driver code
int main()
{
    float l = 5, b = 12, h = 13;
    cout << Area(l, b, h) << 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.*;
import java.text.DecimalFormat;
 
class GFG
{
 
// Function to find the biggest reuleaux triangle
static double Area(double l, double b, double h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    double x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    double A = 0.70477 * Math.pow(x, 2);
 
    return A;
}
 
// Driver code
public static void main(String args[])
{
    double l = 5, b = 12, h = 13;
    DecimalFormat df = new DecimalFormat("#,###,##0.00000");
    System.out.println(df.format(Area(l, b, h)));
}
}
 
// This code is contributed by
// Shashank_Sharma

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 biggest
# reuleaux triangle
def Area(l, b, h):
 
    # the height or base or hypotenuse
    # cannot be negative
    if (l < 0 or b < 0 or h < 0):
        return -1
 
    # height of the reuleaux triangle
    x = (l * b) /(l + b)
 
    # area of the reuleaux triangle
    A = 0.70477 * pow(x, 2)
 
    return A
 
# Driver code
l, b, h = 5, 12, 13
print(Area(l, b, h))
 
# 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 biggest reuleaux triangle
static double Area(double l, double b, double h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    double x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    double A = 0.70477 * Math.Pow(x, 2);
 
    return A;
}
 
// Driver code
public static void Main()
{
    double l = 5, b = 12, h = 13;
    Console.WriteLine((Area(l, b, h)));
}
}
 
// This code is contributed by
// Mukul Singh

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 biggest
// reuleaux triangle
function Area($l, $b, $h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if ($l < 0 or $b < 0 or $h < 0)
        return -1;
 
    // height of the reuleaux triangle
    $x = ($l * $b) / ($l + $b);
 
    // area of the reuleaux triangle
    $A = 0.70477 * pow($x, 2);
 
    return $A;
}
 
// Driver code
$l = 5; $b = 12; $h = 13;
echo Area($l, $b, $h);
 
// This code is contributed by
// anuj_67
?>

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 biggest reuleaux triangle
function Area(l,b,h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    let x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    let A = 0.70477 * Math.pow(x, 2);
 
    return A;
}
 
// Driver code
let l = 5, b = 12, h = 13;
  
    document.write( Area(l,b,h).toFixed(5));
 
// This code contributed by Rajput-Ji
 
</script>
Output: 
8.77914

 

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