Largest rectangle that can be inscribed in a semicircle

Given a semicircle of radius r, we have to find the largest rectangle that can be inscribed in the semicircle, with base lying on the diameter.
Examples:

Input : r = 4
Output : 16

Input : r = 5 
Output :25

Let r be the radius of the semicircle, x one half of the base of the rectangle, and y the height of the rectangle. We want to maximize the area, A = 2xy. 
So from the diagram we have, 
y = ?(r^2 – x^2)
So, A = 2*x*(?(r^2 – x^2)), or dA/dx = 2*?(r^2 – x^2) -2*x^2/?(r^2 – x^2)
Setting this derivative equal to 0 and solving for x, 
dA/dx = 0 
or, 2*?(r^2 – x^2) – 2*x^2/?(r^2 – x^2) = 0
2r^2 – 4x^2 = 0 
x = r/?2
This is the maximum of the area as, 
dA/dx > 0 when x > r/?2
and, dA/dx < 0 when x > r/?2
Since y =?(r^2 – x^2) we then have
y = r/?2
Thus, the base of the rectangle has length = r/?2 and its height has length ?2*r/2. 
So, Area, A=r^2

C++

// C++ Program to find the
// the biggest rectangle
// which can be inscribed
// within the semicircle
#include <bits/stdc++.h>
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

PHP

<?php
// PHP 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
    $a = $r * $r;

    return $a;
}

// Driver code
$r = 5;
echo rectanglearea($r)."\n";

// This code is contributed
// by ChitraNayal
?>

Javascript

<script>

// 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

</script>

OUTPUT :

25

Time Complexity: O(1)
Auxiliary Space: O(1)