# 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; ` `} ` |

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## 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 ` |

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## 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 ` |

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## 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 ` |

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## 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 ` `?> ` |

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**OUTPUT :**

25

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