# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution. ## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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

 ` `

OUTPUT :

```25
```

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