Given a n-sided polygon with side length a. The task is to find the area of the circumcircle of the polygon.
Examples:
Input: n = 10, a = 3 Output: 1.99737
Input: n = 5, a = 6 Output: 3.02487
Approach: A regular n-gon divides the circle into n pieces, so the central angle of the triangle is a full circle divided by n: 360 deg/n.
Applying the law of cosines for the three side lengths of the triangle, we get
c2 = a2 + b2 - 2ab cos C or, a2 = r2 + r2 - 2rr cos (360/n) or, a2 = 2r2 - 2r2 cos (360/n) or, c2 = r2 (2 - 2 cos (360/n)) so, a=r?(2-2cos(360/n)) Therefore, r=a/?(2-2cos(360/n))
Below is the implementation of the above approach:
C++
// C++ Program to find the radius // of the circumcircle of the given polygon #include <bits/stdc++.h> using namespace std;
// Function to find the radius // of the circumcircle float findRadiusOfcircumcircle( float n, float a)
{ // these cannot be negative
if (n < 0 || a < 0)
return -1;
// Radius of the circumcircle
float radius = a / sqrt (2 - (2 * cos (360 / n)));
// Return the radius
return radius;
} // Driver code int main()
{ float n = 5, a = 6;
// Find the radius of the circumcircle
cout << findRadiusOfcircumcircle(n, a) << endl;
return 0;
} |
Java
// Java Program to find the radius // of the circumcircle of the given polygon import java.io.*;
class GFG {
// Function to find the radius // of the circumcircle static float findRadiusOfcircumcircle( float n, float a)
{ // these cannot be negative
if (n < 0 || a < 0 )
return - 1 ;
// Radius of the circumcircle
float radius = ( float )(a / Math.sqrt( 2 - ( 2 * Math.cos( 360 / n))));
// Return the radius
return radius;
} // Driver code public static void main (String[] args) {
float n = 5 , a = 6 ;
// Find the radius of the circumcircle
System.out.println( findRadiusOfcircumcircle(n, a)) ;
}
} // This code is contributed // by anuj_67.. |
Python 3
# Python3 Program to find the # radius of the circumcircle # of the given polygon # from math import all methods from math import *
# Function to find the radius # of the circumcircle def findRadiusOfcircumcircle(n, a) :
# these cannot be negative
if n < 0 or a < 0 :
return - 1
# Radius of the circumcircle
radius = a / sqrt( 2 - ( 2 * cos( 360 / n)))
# Return the radius
return radius
# Driver code if __name__ = = "__main__" :
n , a = 5 , 6
# Find the radius of the circumcircle
print ( round (findRadiusOfcircumcircle(n, a), 5 ))
# This code is contributed # by ANKITRAI1 |
C#
// C# Program to find the radius // of the circumcircle of the given polygon using System;
class GFG
{ // Function to find the radius // of the circumcircle static float findRadiusOfcircumcircle( float n,
float a)
{ // these cannot be negative
if (n < 0 || a < 0)
return -1;
// Radius of the circumcircle
float radius = ( float )(a / Math.Sqrt(2 -
(2 * Math.Cos(360 / n))));
// Return the radius
return radius;
} // Driver code public static void Main ()
{ float n = 5, a = 6;
// Find the radius of the circumcircle
Console.WriteLine(findRadiusOfcircumcircle(n, a));
} } // This code is contributed // by anuj_67 |
PHP
<?php // PHP Program to find the radius // of the circumcircle of the // given polygon // Function to find the radius // of the circumcircle function findRadiusOfcircumcircle( $n , $a )
{ // these cannot be negative
if ( $n < 0 || $a < 0)
return -1;
// Radius of the circumcircle
$radius = $a / sqrt(2 - (2 *
cos (360 / $n )));
// Return the radius
return $radius ;
} // Driver code $n = 5;
$a = 6;
// Find the radius of the circumcircle echo findRadiusOfcircumcircle( $n , $a );
// This code is contributed by Anuj_67.. ?> |
Javascript
<script> // javascript Program to find the radius // of the circumcircle of the given polygon // Function to find the radius // of the circumcircle function findRadiusOfcircumcircle(n , a)
{ // these cannot be negative
if (n < 0 || a < 0)
return -1;
// Radius of the circumcircle
var radius = (a / Math.sqrt(2 - (2 * Math.cos(360 / n))));
// Return the radius
return radius;
} // Driver code var n = 5, a = 6;
// Find the radius of the circumcircle document.write( findRadiusOfcircumcircle(n, a).toFixed(5)) ; // This code is contributed by shikhasingrajput </script> |
Output:
3.02487
Time Complexity : O(log(n)) because it is using inbuilt sqrt function
Auxiliary Space: O(1)