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Radii of the three tangent circles of equal radius which are inscribed within a circle of given radius

Last Updated : 11 Sep, 2023
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Given here is a circle of a given radius. Inside it, three tangent circles of equal radius are inscribed. The task is to find the radii of these tangent circles.

Examples: 

Input: R = 4
Output: 1.858

Input: R = 11
Output: 5.1095

Approach

Let the radii of the tangent circles be r, and the radius of the circumscribing circle 
is R.x is the smaller distance from the circumference of the tangent circle and the center of the circumscribing circle.

From the diagram, it is very clear, 
2r + x = R

  • now in triangle OBC
    cos 30 = r/(r+x) 
    rcos30 + xcos30 = r 
    x = r(1-cos30)/cos30
  • also, x = R-2r
  • So, 
    R-2r = r(1-cos30)/cos30 
    R-2r = 0.133r/0.867 
    R-2r = 0.153r 
    R = 2.153r 
    so, r = 0.4645R 

Radii of three tangent circles = 0.4645*radius of the circumscribed circle

Below is the implementation of the above approach:

C++




// C++ program to find the radii
// of the three tangent circles
// of equal radius when the radius
// of the circumscribed circle is given
 
#include <bits/stdc++.h>
using namespace std;
 
void threetangcircle(int R)
{
    cout << "The radii of the tangent circles is "
         << 0.4645 * R << endl;
}
 
// Driver code
int main()
{
    int R = 4;
    threetangcircle(R);
    return 0;
}


Java




// Java program to find the radii
// of the three tangent circles
// of equal radius when the radius
// of the circumscribed circle is given
import java.io.*;
 
class GFG {
 
    static void threetangcircle(int R)
    {
        System.out.print(
            "The radii of the tangent circles is "
            + 0.4645 * R);
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int R = 4;
        threetangcircle(R);
    }
}
 
// This code is contributed by anuj_67..


Python3




# Python3 program to find the radii
# of the three tangent circles
# of equal radius when the radius
# of the circumscribed circle is given
 
 
def threetangcircle(R):
    print("The radii of the tangent",
          "circles is ", end="")
    print(0.4645 * R)
 
 
# Driver code
R = 4
threetangcircle(R)
 
# This code is contributed
# by Princi Singh


C#




// C# program to find the radii
// of the three tangent circles
// of equal radius when the radius
// of the circumscribed circle is given
using System;
 
class GFG {
 
    static void threetangcircle(int R)
    {
        Console.WriteLine(
            "The radii of the tangent circles is "
            + 0.4645 * R);
    }
 
    // Driver code
    public static void Main()
    {
        int R = 4;
        threetangcircle(R);
    }
}
 
// This code is contributed by anuj_67..


Javascript




<script>
// javascript program to find the radii
// of the three tangent circles
// of equal radius when the radius
// of the circumscribed circle is given
 
function threetangcircle(R)
{
    document.write("The radii of the tangent circles is "
        + 0.4645 * R);
}
 
// Driver code
var R = 4;
threetangcircle(R);
 
 
// This code contributed by shikhasingrajput
</script>


PHP




<?php
// PHP program to find the radii
// of the three tangent circles
// of equal radius when the radius
// of the circumscribed circle is given
 
function threetangcircle($R)
{
    echo "The radii of the tangent circles is ",
        ( 0.4645 * $R );
}
 
// Driver code
$R = 4;
threetangcircle($R);
     
// This code is contributed by ihritik
?>


Output

The radii of the tangent circles is 1.858

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



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