最大的鲁洛三角形内切于半圆内切的正方形内

这里给出的是一个半径为r的半圆，它内接一个正方形，而正方形又内接一个鲁洛三角形。任务是找到这个鲁洛三角形的最大可能面积。
例子：

Input : x = 5  
Output : 14.0954

Input : x = 8
Output : 36.0842

求法：我们知道，内接半圆的正方形的边长是a = 2r/√5 。 (请参考这里)
此外，在鲁洛三角形中， x = a 。
所以， x = 2*r/√5
所以，鲁洛三角形的面积：

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

下面是上述方法的实现：

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

Javascript

// javascript 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
    var x = (2 * r) /(Math.sqrt(5));
 
    // area of the reuleaux triangle
    var A = 0.70477 *(Math.pow(x, 2));
    return A;
}
 
// Driver code
var r = 5;
document.write(Area(r).toFixed(4));
 
 
// This code is contributed by Princi Singh

输出：

14.0954