Circumradius of a Cyclic Quadrilateral using the length of Sides

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Given integers A, B, C, and D denoting the length of sides of a Cyclic Quadrilateral, the task is to find the circumradius i.e., the radius of circumcircle of the given cyclic quadrilateral.

Examples:

Input: A = 3, B = 4, C = 5, D= 6
Output: 3.29

Input: A = 10, B = 30, C = 50, D = 20
Output: 27.78

Approach: Follow the steps below to solve the problem:

Calculate the semiperimeter of the cyclic quadrilateral with sides A, B, C and D by using the equation:

$Semiperimeter(s)=\frac{a+b+c+d}{2}$

Now, using Parameshvara’s circumradius formula shown below, calculate the radius for the circumcircle:

$R = \frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}$

Below is the implementation of the above approach:

C++14

// C++ program to find circumradius of
// a cyclic quadrilateral using sides
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the circumradius
// of a cyclic quadrilateral using sides
double Circumradius(int a, int b, int c, int d)
{
     
    // Find semiperimeter
    double s = (a + b + c + d) / 2.0;
 
    // Calculate the radius
    double radius = sqrt(((a * b) + (c * d)) *
                         ((a * c) + (b * d)) * 
                         ((a * d) + (b * c)) / 
                         ((s - a) * (s - b) * 
                          (s - c) * (s - d)));
 
    return radius / 4;
}
 
// Driver Code
int main()
{
    int A = 3;
    int B = 4;
    int C = 5;
    int D = 6;
 
    // Function call
    double ans = Circumradius(A, B, C, D);
 
    // Print the radius
    cout << setprecision(3) << ans;
 
    return 0;
}
 
// This code is contributed by mohit kumar 29

Java

// Java program to find circumradius of
// a cyclic quadrilateral using sides
import java.util.*;
 
class GFG{
 
// Function to return the circumradius
// of a cyclic quadrilateral using sides
static double Circumradius(int a, int b, 
                           int c, int d)
{
     
    // Find semiperimeter
    double s = (a + b + c + d) / 2.0;
 
    // Calculate the radius
    double radius = Math.sqrt(((a * b) + (c * d)) *
                              ((a * c) + (b * d)) * 
                              ((a * d) + (b * c)) / 
                              ((s - a) * (s - b) * 
                               (s - c) * (s - d)));
 
    return radius / 4;
}
 
// Driver Code
public static void main(String[] args)
{
    int A = 3;
    int B = 4;
    int C = 5;
    int D = 6;
 
    // Function call
    double ans = Circumradius(A, B, C, D);
 
    // Print the radius
    System.out.format("%.2f", ans);
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Program to find Circumradius of
# a cyclic quadrilateral using sides
 
import math  
   
# Function to return the Circumradius
# of  a cyclic quadrilateral using sides
def Circumradius(a, b, c, d):
     
# Find semiperimeter
    s = (a + b + c + d) / 2
 
    # Calculate the radius
    radius = (1 / 4)*math.sqrt(((a * b)+(c * d))*
        ((a * c)+(b * d))*((a * d)+(b * c))
        /((s-a)*(s-b)*(s-c)*(s-d)))
     
    return radius  
   
# Driver Code
 
# Given sides
A = 3
B = 4
C = 5
D = 6
   
# Function Call  
ans = Circumradius(A, B, C, D)  
   
# Print the radius
print(round(ans, 2))

C#

// C# program to find circumradius of
// a cyclic quadrilateral using sides
using System;
 
class GFG{
 
// Function to return the circumradius
// of a cyclic quadrilateral using sides
static double Circumradius(int a, int b, 
                           int c, int d)
{
     
    // Find semiperimeter
    double s = (a + b + c + d) / 2.0;
 
    // Calculate the radius
    double radius = Math.Sqrt(((a * b) + (c * d)) *
                              ((a * c) + (b * d)) * 
                              ((a * d) + (b * c)) / 
                              ((s - a) * (s - b) * 
                               (s - c) * (s - d)));
 
    return radius / 4;
}
 
// Driver Code
public static void Main(String[] args)
{
    int A = 3;
    int B = 4;
    int C = 5;
    int D = 6;
 
    // Function call
    double ans = Circumradius(A, B, C, D);
 
    // Print the radius
    Console.Write("{0:F2}", ans);
}
}
 
// This code is contributed by 29AjayKumar 

Javascript

<script>
 
// Javascript program to find circumradius of
// a cyclic quadrilateral using sides
 
// Function to return the circumradius
// of a cyclic quadrilateral using sides
function Circumradius(a, b, c, d)
{
     
    // Find semiperimeter
    var s = (a + b + c + d) / 2.0;
 
    // Calculate the radius
    var radius = Math.sqrt(((a * b) + (c * d)) *
                          ((a * c) + (b * d)) * 
                          ((a * d) + (b * c)) / 
                          ((s - a) * (s - b) * 
                          (s - c) * (s - d)));
 
    return radius / 4;
}
 
// Driver code
var A = 3;
var B = 4;
var C = 5;
var D = 6;
 
// Function call
var ans = Circumradius(A, B, C, D);
 
// Print the radius
document.write(ans.toFixed(2));
     
// This code is contributed by Khushboogoyal499
 
</script>

Output:

3.29

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

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