Maximum area of a Rectangle that can be circumscribed about a given Rectangle of size LxW

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Given a rectangle of dimensions L and W. The task is to find the maximum area of a rectangle that can be circumscribed about a given rectangle with dimensions L and W.

Examples:

Input: L = 10, W = 10
Output: 200

Input: L = 18, W = 12
Output: 450

Approach: Let below is the given rectangle EFGH of dimensions L and W. We have to find the area of rectangle ABCD which is circumscribing rectangle EFGH.

In the above figure:
If $\angle ABC = X$ then $\angle CGF = 90 - X$ as GCF is right angled triangle.
Therefore,
$\angle HGD = 180 - \angle FGH - \angle CGF$
=> $\angle HGD = 180 - 90 - (90 - X)$
=> $\angle HGD = X$
Similarly,
$\angle EHA = X$
$\angle FEB = X$
Now, The area of rectangle ABCD is given by:

Area = AB * AD
Area = (AE + EB)*(AH + HD) …..(1)

According to the projection rule:
AE = L*sin(X)
EB = W*cos(X)
AH = L*cos(X)
HD = W*sin(X)

Substituting the value of the above projections in equation (1) we have:

$Area = (AE + EB)*(AH + HD)$
$Area = (L*sin(X)+ W*cox(X))*(L*cos(X) + W*sin(X))$
$Area = ((L^{2} + W^{2})*sin(X)*cos(X) + L*W)$
$Area = (\frac{(L^{2} + W^{2})*sin(2X)}{2} + L*W)$
Now to maximize the area, the value of sin(2X) must be maximum i.e., 1.
Therefore after substituting sin(2X) as 1 we have,
$Area = (\frac{(L^{2} + W^{2})}{2} + L*W)$
$Area = (\frac{(L+W)^{2}}{2})$

Below is the implementation of the above approach:

C++

// C++ program for the above approach 
#include <bits/stdc++.h> 
using namespace std; 
 
// Function to find area of rectangle 
// inscribed another rectangle of 
// length L and width W 
double AreaofRectangle(int L, int W)
{
     
    // Area of rectangle 
    double area = (W + L) * (W + L) / 2;
     
    // Return the area 
    return area;
}
 
// Driver Code 
int main() 
{ 
     
    // Given dimensions 
    int L = 18;
    int W = 12;
     
    // Function call 
    cout << AreaofRectangle(L, W);
    return 0; 
} 
 
// This code is contributed by Princi Singh

Java

// Java program for the above approach 
import java.io.*;
import java.util.*; 
 
class GFG{
     
// Function to find area of rectangle 
// inscribed another rectangle of 
// length L and width W 
static double AreaofRectangle(int L, int W)
{
     
    // Area of rectangle 
    double area = (W + L) * (W + L) / 2;
     
    // Return the area 
    return area;
}
     
// Driver Code 
public static void main(String args[])
{ 
     
    // Given dimensions 
    int L = 18;
    int W = 12;
     
    // Function call 
    System.out.println(AreaofRectangle(L, W));
} 
} 
 
// This code is contributed by offbeat

Python3

# Python3 program for the above approach
 
# Function to find area of rectangle 
# inscribed another rectangle of 
# length L and width W
def AreaofRectangle(L, W):
   
  # Area of rectangle
  area =(W + L)*(W + L)/2
 
# Return the area
  return area
 
# Driver Code
if __name__ == "__main__":
 
  # Given Dimensions
  L = 18
  W = 12
 
  # Function Call
  print(AreaofRectangle(L, W))

C#

// C# program for the above approach 
using System;
 
class GFG{
     
// Function to find area of rectangle 
// inscribed another rectangle of 
// length L and width W 
static double AreaofRectangle(int L, int W)
{
     
    // Area of rectangle 
    double area = (W + L) * (W + L) / 2;
     
    // Return the area 
    return area;
}
     
// Driver Code 
public static void Main(String []args)
{ 
     
    // Given dimensions 
    int L = 18;
    int W = 12;
     
    // Function call 
    Console.Write(AreaofRectangle(L, W));
} 
} 
 
// This code is contributed by shivanisinghss2110

Javascript

<script>
      // JavaScript program for the above approach
 
      // Function to find area of rectangle
      // inscribed another rectangle of
      // length L and width W
      function AreaofRectangle(L, W) {
        // Area of rectangle
        var area = parseFloat(((W + L) * (W + L)) / 2).toFixed(1);
 
        // Return the area
        return area;
      }
 
      // Driver Code
      // Given dimensions
      var L = 18;
      var W = 12;
 
      // Function call
      document.write(AreaofRectangle(L, W));
    </script>

Output:

450.0

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

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