# 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

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

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++

 // C++ Program to find the biggest Reuleaux triangle // inscribed within in a square which in turn // is inscribed within a semicircle    #include 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; }

## Java

 // 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.

## Python3

 # 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

## C#

 // 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

## PHP



Output:

14.0954

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