Area of circle inscribed in a Isosceles Trapezoid

Last Updated : 07 Sep, 2021

Given two bases of the isosceles trapezoid ABCD as a and b, the task is to find the area of circle inscribed in this trapezoid

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Examples:

Input: a = 10, b = 30 
Output: Area = 235.57

Input: a = 20, b = 36 
Output: Area = 565.38

Derivation: Given a circle inscribed in trapezium ABCD (sides AB = n and CD = m), we need to find out the height of the trapezium i.e., (AL), which is half of the radius of the circle to find the area of the circle.

For finding the height of circle we do following operation.

The circle will always touch the sides of trapezium at their midpoints, Say the midpoints of AB, BD, CD, AC are G, F, H, E, and join them with the center of the circle.

Now from Symmetry, we can see that

AG = AE = n/2, 
EC = CG = m/2, 
HD = DF = n/2,
GB = FB = m/2

Now in Triangle ACL apply the Pythagoras theorem.

Hypotenuse AC = m/2 + n/2
Base CL = CH - AG = m/2 - n/2

we get 
Perpendicular AL = Square_root(m * n)

Therefore the height of the Trapezium = AL = Square_Root(Product of given sides)
Now the radius of the circle is simple half of the height and hence the area can be calculated easily.

Approach:

Find the height of the trapezoid as (square_root( m * n )).

Find the radius of the incircle

R = height / 2 
  = square_root(m * n) / 2

Now find the area of the circle
```
= Pi * R² 
= ( 3.141 * m * n ) / 4
```

Below is the implementation of the above approach:

C++

// CPP implementation to find
// the rea of the circle
// inscribed in a trapezoid
// having non- parallel sides m, n
#include<bits/stdc++.h>
using namespace std;
  
// Function to find area of circle
// inscribed in a trapezoid
// having non- parallel sides m, n
double area_of_circle(int m, int n)
{
    // radius of circle by the
    // formula i.e. root( m * n) / 2
    // area of circle = (3.141 ) * ( R ** 2 )
  
    int square_of_radius = ( m * n ) / 4;
    double area = ( 3.141 * square_of_radius );
    return area;
}
  
// Driver Code
int main(){
    int n = 10;
    int m = 30;
    cout << (area_of_circle(m, n));
}
  
// This code is contributed by mohit kumar 29

Java

// Java Program to find 
// the rea of the circle 
// inscribed in a trapezoid 
// having non- parallel sides m, n 
class GFG 
{ 
      
    // Function to find area of circle 
    // inscribed in a trapezoid 
    // having non- parallel sides m, n 
    static double area_of_circle(int m, int n) 
    { 
        // radius of circle by the 
        // formula i.e. root( m * n) / 2 
        // area of circle = (3.141 ) * ( R ** 2 ) 
      
        int square_of_radius = ( m * n ) / 4; 
        double area = ( 3.141 * square_of_radius ); 
        return area; 
    } 
      
    // Driver code 
    public static void main (String[] args) 
    { 
        int n = 10; 
        int m = 30; 
        System.out.println(area_of_circle(m, n)); 
    } 
} 
  
// This code is contributed by Yash_R

Python3

# Python 3 implementation to find 
# the rea of the circle 
# inscribed in a trapezoid
# having non- parallel sides  m, n
  
  
# Function to find area of circle 
# inscribed in a trapezoid
# having non- parallel sides  m, n
def area_of_circle(m, n):
    # radius of circle by the
    # formula i.e. root( m * n) / 2
    # area of circle = (3.141 ) * ( R ** 2 )
      
    square_of_radius = ( m * n ) / 4
    area = ( 3.141 * square_of_radius )
    return area
  
# Driver Code 
if __name__=='__main__': 
    n = 10
    m = 30
    print(area_of_circle(m, n))

C#

// C# Program to find 
// the rea of the circle 
// inscribed in a trapezoid 
// having non- parallel sides m, n 
using System; 
  
class GFG 
{ 
      
// Function to find area of circle 
// inscribed in a trapezoid 
// having non- parallel sides m, n 
static double area_of_circle(int m, int n) 
{ 
    // radius of circle by the 
    // formula i.e. root( m * n) / 2 
    // area of circle = (3.141 ) * ( R ** 2 ) 
  
    int square_of_radius = ( m * n ) / 4; 
    double area = ( 3.141 * square_of_radius ); 
    return area; 
} 
  
// Driver code 
public static void Main () 
{ 
    int n = 10; 
    int m = 30;
    Console.WriteLine(area_of_circle(m, n)); 
} 
} 
  
// This code is contributed by Sanjit_Prasad

Javascript

<script>
  
// Javascript Program to find 
// the rea of the circle 
// inscribed in a trapezoid 
// having non- parallel sides m, n 
  
// Function to find area of circle 
// inscribed in a trapezoid 
// having non- parallel sides m, n 
function area_of_circle(m, n) 
{ 
      
    // Radius of circle by the 
    // formula i.e. root( m * n) / 2 
    // area of circle = (3.141 ) * ( R ** 2 ) 
  
    var square_of_radius = ( m * n ) / 4; 
    var area = ( 3.141 * square_of_radius ); 
    return area; 
} 
  
// Driver Code 
var n = 10; 
var m = 30; 
  
document.write(area_of_circle(m, n)); 
  
// This code is contributed by Khushboogoyal499
      
</script>

Output:

235.575

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Area of circle inscribed in a Isosceles Trapezoid

C++

Java

Python3

C#

Javascript

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