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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 : 04 Apr, 2019

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++

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// 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;
}

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Java

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// 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));
    }
}

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Python3

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# 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))

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C#

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// 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

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PHP

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<?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
?>

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Output:

57

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