Angle of intersection of two circles having their centers D distance apart

Given two positive integers R1 and R2 representing the radius of two intersecting circles having a distance D between their centers, the task is to find the cosine of the angle of intersection between the two circles.

Examples:

Input: R1 = 3, R2 = 4, D = 5
Output: 0

Input: R1 = 7, R2 = 3, D = 6
Output: 0.52381

Approach: The given problem can be solved by using the Geometric Algorithm as illustrated below:

From the above image and using the Pythagoras Theorem, the cosine of the angle of intersection of the two circles can be found using the formula:

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the cosine of the
// angle of the intersection of two
// circles with radius R1 and R2
float angle(float R1, float R2, float D)
{
    float ans = (R1 * R1 + R2 * R2 - D * D)
                / (2 * R1 * R2);

    // Return the cosine of the angle
    return ans;
}

// Driver Code
int main()
{
    float R1 = 3, R2 = 4;
    float D = 5;
    cout << angle(R1, R2, D);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG{

// Function to find the cosine of the
// angle of the intersection of two
// circles with radius R1 and R2
static float angle(float R1, float R2, float D)
{
    float ans = (R1 * R1 + R2 * R2 - D * D) /
               (2 * R1 * R2);

    // Return the cosine of the angle
    return ans;
}

// Driver Code
public static void main (String[] args)
{
    float R1 = 3, R2 = 4;
    float D = 5;

    System.out.println(angle(R1, R2, D));
}
}

// This code is contributed by Ankita saini

Python3

# Python3 program for the above approach

# Function to find the cosine of the
# angle of the intersection of two
# circles with radius R1 and R2
def angle(R1, R2, D):

    ans = ((R1 * R1 + R2 * R2 - D * D) /
            (2 * R1 * R2))

    # Return the cosine of the angle
    return ans

# Driver Code
if __name__ == '__main__':

    R1 = 3
    R2 = 4
    D = 5

    print(angle(R1, R2, D))

# This code is contributed by ipg2016107

C#

// C# program for the above approach
using System;

class GFG{

// Function to find the cosine of the
// angle of the intersection of two
// circles with radius R1 and R2
static float angle(float R1, float R2, float D)
{
    float ans = (R1 * R1 + R2 * R2 - D * D) /
               (2 * R1 * R2);

    // Return the cosine of the angle
    return ans;
}

// Driver Code
public static void Main(string[] args)
{
    float R1 = 3, R2 = 4;
    float D = 5;

    Console.Write(angle(R1, R2, D));
}
}

// This code is contributed by rutvik_56.

Javascript

<script>

// Javascript program for the above approach

// Function to find the cosine of the
// angle of the intersection of two
// circles with radius R1 and R2
function angle(R1, R2, D)
{
    var ans = (R1 * R1 + R2 * R2 - D * D) /
               (2 * R1 * R2);

    // Return the cosine of the angle
    return ans;
}

// Driver Code
var R1 = 3, R2 = 4;
var D = 5;

document.write(angle(R1, R2, D));

// This code is contributed by Ankita saini

</script>

Output:

0

Time Complexity: O(1)
Auxiliary Space: O(1)