可以刻在半圆上的最大矩形

给定半径为r的半圆，我们必须找到可以在该半圆中内接的最大矩形，其底数位于直径上。
例子：

Input : r = 4
Output : 16

Input : r = 5 
Output :25

令r为半圆的半径， x为矩形底边的一半， y为矩形的高度。我们要最大化面积，A = 2xy。
因此，从图中我们可以看到，
y =√(r ^ 2 – x ^ 2)
因此， A = 2 * x *(√(r ^ 2 – x ^ 2))或dA / dx = 2 *√(r ^ 2 – x ^ 2)-2 * x ^ 2 /√(r ^ 2 – x ^ 2)
将此导数设置为0并求解x，
dA / dx = 0
或2 *√(r ^ 2 – x ^ 2)– 2 * x ^ 2 /√(r ^ 2 – x ^ 2)= 0
2r ^ 2 – 4x ^ 2 = 0
x = r /√2
这是该区域的最大值，因为
当x> r /√2时dA / dx> 0
并且，当x> r /√2时dA / dx <0
由于y =√(r ^ 2 – x ^ 2)
y = r /√2
因此，矩形的底部的长度为r /√2 ，其高度的长度为√2* r / 2 。
因此，面积， A = r ^ 2

C++

// C++ Program to find the
// the biggest rectangle
// which can be inscribed
// within the semicircle
#include 
using namespace std;
 
// Function to find the area
// of the biggest rectangle
float rectanglearea(float r)
{
 
    // the radius cannot be negative
    if (r < 0)
        return -1;
 
    // area of the rectangle
    float a = r * r;
 
    return a;
}
 
// Driver code
int main()
{
    float r = 5;
    cout << rectanglearea(r) << endl;
    return 0;
}

Java

// Java Program to find the
// the biggest rectangle
// which can be inscribed
// within the semicircle
class GFG
{
 
// Function to find the area
// of the biggest rectangle
static float rectanglearea(float r)
{
 
// the radius cannot be negative
if (r < 0)
    return -1;
 
// area of the rectangle
float a = r * r;
 
return a;
}
 
// Driver code
public static void main(String[] args)
{
    float r = 5;
    System.out.println((int)rectanglearea(r));
}
}
 
// This code is contributed
// by ChitraNayal

Python 3

# Python 3 Program to find the
# the biggest rectangle
# which can be inscribed
# within the semicircle
 
# Function to find the area
# of the biggest rectangle
def rectanglearea(r) :
 
    # the radius cannot
    # be negative
    if r < 0 :
        return -1
 
    # area of the rectangle
    a = r * r
 
    return a
 
# Driver Code
if __name__ == "__main__" :
 
    r = 5
 
    # function calling
    print(rectanglearea(r))
 
# This code is contributed
# by ANKITRAI1

C#

// C# Program to find the
// the biggest rectangle
// which can be inscribed
// within the semicircle
using System;
 
class GFG
{
 
// Function to find the area
// of the biggest rectangle
static float rectanglearea(float r)
{
 
// the radius cannot be negative
if (r < 0)
    return -1;
 
// area of the rectangle
float a = r * r;
 
return a;
}
 
// Driver code
public static void Main()
{
    float r = 5;
    Console.Write((int)rectanglearea(r));
}
}
 
// This code is contributed
// by ChitraNayal

Javascript

 
// javascript Program to find the
// the biggest rectangle
// which can be inscribed
// within the semicircle
 
// Function to find the area
// of the biggest rectangle
function rectanglearea(r)
{
 
    // the radius cannot be negative
    if (r < 0)
        return -1;
     
    // area of the rectangle
    var a = r * r;
     
    return a;
}
 
// Driver code
 
var r = 5;
document.write(parseInt(rectanglearea(r)));
 
// This code is contributed by Amit Katiyar
 

输出 ：

25