Largest square that can be inscribed in a semicircle

Given a semicircle with radius r, we have to find the largest square that can be inscribed in the semicircle, with base lying on the diameter.

Examples:

Input: r = 5
Output: 20

Input: r = 8
Output: 51.2


Approach: Let r be the radius of the semicircle & a be the side length of the square.
From the figure we can see that, centre of the circle is also the midpoint of the base of the square.So in the right angled triangle AOB, from Pythagorus Theorem:

a^2 + (a/2)^2 = r^2

5*(a^2/4) = r^2

a^2 = 4*(r^2/5) i.e. area of the square

Below is the implementation of the above approach:

C++

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// C++ Program to find the the biggest square
// which can be inscribed within the semicircle
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the area
// of the squaare
float squarearea(float r)
{
  
    // the radius cannot be negative
    if (r < 0)
        return -1;
  
    // area of the square
    float a = 4 * (pow(r, 2) / 5);
  
    return a;
}
  
// Driver code
int main()
{
    float r = 5;
    cout << squarearea(r) << endl;
  
    return 0;
}

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Java

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// Java Program to find the the biggest square
// which can be inscribed within the semicircle
  
import java.io.*;
  
class GFG {
  
  
// Function to find the area
// of the squaare
static float squarearea(float r)
{
  
    // the radius cannot be negative
    if (r < 0)
        return -1;
  
    // area of the square
    float a = 4 * (float)(Math.pow(r, 2) / 5);
  
    return a;
}
  
// Driver code
  
    public static void main (String[] args) {
         float r = 5;
    System.out.println( squarearea(r));
    }
}
// This code is contributed by chandan_jnu.

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Python3

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# Python 3 program to find the the 
# biggest square which can be 
# inscribed within the semicircle
  
# Function to find the area
# of the squaare
def squarearea(r):
  
    # the radius cannot be
    # negative
    if (r < 0):
        return -1
  
    # area of the square
    a = 4 * (pow(r, 2) / 5)
  
    return a
  
# Driver code
if __name__ == "__main__":
      
    r = 5
    print(int(squarearea(r)))
  
# This code is contributed
# by ChitraNayal

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C#

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// C# Program to find the the 
// biggest square which can be
// inscribed within the semicircle
using System;
  
class GFG
{
  
// Function to find the 
// area of the squaare
static float squarearea(float r)
{
  
    // the radius cannot be negative
    if (r < 0)
        return -1;
  
    // area of the square
    float a = 4 * (float)(Math.Pow(r, 2) / 5);
  
    return a;
}
  
// Driver code
public static void Main ()
{
    float r = 5;
    Console.WriteLine(squarearea(r));
}
}
  
// This code is contributed 
// by anuj_67

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PHP

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<?php
// PHP Program to find the the 
// biggest square which can be 
// inscribed within the semicircle
  
// Function to find the area
// of the squaare
function squarearea($r)
{
  
    // the radius cannot be negative
    if ($r < 0)
        return -1;
  
    // area of the square
    $a = 4 * (pow($r, 2) / 5);
  
    return $a;
}
  
// Driver code
$r = 5;
echo squarearea($r);
  
// This code is contributed 
// by Shivi_Aggarwal
?>

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Output:

20


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