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Largest rectangle that can be inscribed in a semicircle
  • Last Updated : 15 Mar, 2021

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

 

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