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Find the area of the shaded region formed by the intersection of four semicircles in a square

  • Difficulty Level : Medium
  • Last Updated : 19 Apr, 2021

Given the length of the side of a square a, the task is to find the area of the shaded region formed by the intersection of four semicircles in a square as shown in the image below: 
 

Examples: 
 

Input: a = 10 
Output: 57
Input: a = 19 
Output: 205.77 
 

 



Approach: Area of the shaded region will be: 
 

Area(semicircle1) + Area(semicircle2) + Area(semicircle3) + Area(semicircle4) – Area(square).
Since all semicircles are of same radius, therefore, area of all semicircles will be equal. So, the above formula can be written as: 
4*(Area of Semicircle) – Area(Square) 
 

The area of a semicircle is (3.14 * r2) / 2 where r is the radius of the semicircle which is equal to a / 2.
Hence, Area of the shaded region = 4 * (3.14 * (a * a) / 8 ) – a * a
 

Below is the implementation of the above approach:
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the area
// of the shaded region
float findAreaShaded(float a)
{
 
    // Area of the square
    float sqArea = a * a;
 
    // Area of the semicircle
    float semiCircleArea = (3.14 * (a * a) / 8);
 
    // There are 4 semicircles
    // shadedArea = Area of 4 semicircles - Area of square
    float ShadedArea = 4 * semiCircleArea - sqArea;
 
    return ShadedArea;
}
 
// Driver code
int main()
{
    float a = 10;
    cout << findAreaShaded(a);
 
    return 0;
}

Java




// Java implementation of the approach
class GFG {
 
    // Function to return the area
    // of the shaded region
    static float findAreaShaded(float a)
    {
 
        // Area of the square
        float sqArea = a * a;
 
        // Area of the semicircle
        float semiCircleArea = (float)(3.14 * (a * a) / 8);
 
        // There are 4 semicircles
        // shadedArea = Area of 4 semicircles - Area of square
        float ShadedArea = 4 * semiCircleArea - sqArea;
 
        return ShadedArea;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        float a = 10;
        System.out.println(findAreaShaded(a));
    }
}

Python3




# Python3 implementation of the approach
 
# Function to return the area
# of the shaded region
def findAreaShaded(a):
     
    # Area of the square
    sqArea = a * a;
 
    # Area of the semicircle
    semiCircleArea = (3.14 * (a * a ) / 8)
 
    # There are 4 semicircles
    # shadedArea = Area of 4 semicircles - Area of square
    ShadedArea = 4 * semiCircleArea - sqArea ;
 
    return ShadedArea;
 
# Driver code
if __name__ == '__main__':
    a = 10
    print(findAreaShaded(a))

C#




// C# implementation of the approach
using System;
 
class GFG
{
 
    // Function to return the area
    // of the shaded region
    static float findAreaShaded(float a)
    {
 
        // Area of the square
        float sqArea = a * a;
 
        // Area of the semicircle
        float semiCircleArea = (float)(3.14 * (a * a) / 8);
 
        // There are 4 semicircles
        // shadedArea = Area of 4 semicircles - Area of square
        float ShadedArea = 4 * semiCircleArea - sqArea;
 
        return ShadedArea;
    }
 
    // Driver code
    public static void Main()
    {
        float a = 10;
        Console.WriteLine(findAreaShaded(a));
    }
}
 
// This code is contributed by mohit kumar 29

PHP




<?php
// PHP implementation of the approach
 
// Function to return the area
// of the shaded region
function findAreaShaded($a)
{
 
    // Area of the square
    $sqArea = $a * $a;
 
    // Area of the semicircle
    $semiCircleArea = (3.14 * ($a * $a) / 8);
 
    // There are 4 semicircles
    // shadedArea = Area of 4 semicircles -
    //              Area of square
    $ShadedArea = 4 * $semiCircleArea - $sqArea;
 
    return $ShadedArea;
}
 
// Driver code
$a = 10;
echo findAreaShaded($a);
 
// This code is contributed by Ryuga
?>

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function to return the area
// of the shaded region
function findAreaShaded( a)
{
 
    // Area of the square
    let sqArea = a * a;
 
    // Area of the semicircle
    let semiCircleArea = (3.14 * (a * a) / 8);
 
    // There are 4 semicircles
    // shadedArea = Area of 4 semicircles - Area of square
    let ShadedArea = 4 * semiCircleArea - sqArea;
 
    return ShadedArea;
}
 
 
    // driver code
 
    let a = 10;
    document.write(findAreaShaded(a));
     
</script>
Output: 
57

 

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