# 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 ` `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

 ``

## Javascript

 ``

OUTPUT :

`25`

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

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