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.Â
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))Â
Below is the implementation of the above approach:Â
C++
// C++ Program to find the radius// of the circumcircle of the given polygonÂ
#include <bits/stdc++.h>using namespace std;Â
// Function to find the radius// of the circumcirclefloat findRadiusOfcircumcircle(float n, float a){Â
    // these cannot be negative    if (n < 0 || a < 0)        return -1;Â
    // Radius of the circumcircle    float radius = a / sqrt(2 - (2 * cos(360 / n)));Â
    // Return the radius    return radius;}Â
// Driver codeint main(){Â
    float n = 5, a = 6;Â
    // Find the radius of the circumcircle    cout << findRadiusOfcircumcircle(n, a) << endl;Â
    return 0;} |
Java
// Java Program to find the radius// of the circumcircle of the given polygonÂ
import java.io.*;Â
class GFG {Â Â Â // Function to find the radius// of the circumcirclestatic float findRadiusOfcircumcircle(float n, float a){Â
    // these cannot be negative    if (n < 0 || a < 0)        return -1;Â
    // Radius of the circumcircle    float radius = (float)(a / Math.sqrt(2 - (2 * Math.cos(360 / n))));Â
    // Return the radius    return radius;}Â
// Driver codeÂ
    public static void main (String[] args) {        float n = 5, a = 6;Â
    // Find the radius of the circumcircle    System.out.println( findRadiusOfcircumcircle(n, a)) ;Â
    }}// This code is contributed // by anuj_67.. |
Python 3
# Python3 Program to find the # radius of the circumcircle # of the given polygon Â
# from math import all methodsfrom math import *Â
# Function to find the radius # of the circumcircle def findRadiusOfcircumcircle(n, a) :Â
    # these cannot be negative    if n < 0 or a < 0 :        return -1Â
    # Radius of the circumcircle    radius = a / sqrt(2 - (2 * cos(360 / n)))Â
    # Return the radius    return radiusÂ
# Driver codeif __name__ == "__main__" :Â
    n , a = 5, 6Â
    # Find the radius of the circumcircle     print(round(findRadiusOfcircumcircle(n, a), 5))Â
# This code is contributed # by ANKITRAI1 |
C#
// C# Program to find the radius// of the circumcircle of the given polygonusing System;Â
class GFG {Â
// Function to find the radius// of the circumcirclestatic float findRadiusOfcircumcircle(float n, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â float a){Â
    // these cannot be negative    if (n < 0 || a < 0)        return -1;Â
    // Radius of the circumcircle    float radius = (float)(a / Math.Sqrt(2 -                    (2 * Math.Cos(360 / n))));Â
    // Return the radius    return radius;}Â
// Driver codepublic static void Main () {Â Â Â Â float n = 5, a = 6;Â
    // Find the radius of the circumcircle    Console.WriteLine(findRadiusOfcircumcircle(n, a));}}Â
// This code is contributed // by anuj_67 |
PHP
<?php// PHP Program to find the radius // of the circumcircle of the // given polygon Â
// Function to find the radius // of the circumcircle function findRadiusOfcircumcircle($n, $a) { Â
    // these cannot be negative     if ($n < 0 || $a < 0)         return -1; Â
    // Radius of the circumcircle     $radius = $a / sqrt(2 - (2 *                     cos(360 / $n))); Â
    // Return the radius     return $radius; } Â
// Driver code $n = 5;$a = 6; Â
// Find the radius of the circumcircle echo findRadiusOfcircumcircle($n, $a); Â
// This code is contributed by Anuj_67..?> |
Javascript
<script>// javascript Program to find the radius// of the circumcircle of the given polygonÂ
// Function to find the radius// of the circumcirclefunction findRadiusOfcircumcircle(n , a){Â
    // these cannot be negative    if (n < 0 || a < 0)        return -1;Â
    // Radius of the circumcircle    var radius = (a / Math.sqrt(2 - (2 * Math.cos(360 / n))));Â
    // Return the radius    return radius;}Â
// Driver codevar n = 5, a = 6;Â
// Find the radius of the circumcircledocument.write( findRadiusOfcircumcircle(n, a).toFixed(5)) ;Â
Â
// This code is contributed by shikhasingrajput </script> |
Output:Â
3.02487
Â
Time Complexity : O(log(n)) because it is using inbuilt sqrt function
Auxiliary Space: O(1)
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