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# Calculate area of a cyclic quadrilateral with given side lengths

• Difficulty Level : Hard
• Last Updated : 04 Oct, 2021

Given four positive integers A, B, C, and D representing the length of sides of a Cyclic Quadrilateral, the task is to find the area of the Cyclic Quadrilateral.

Examples:

Input: A = 10, B = 15, C = 20, D = 25
Output: 273.861

Input: A = 10, B = 30, C = 50, D = 20
Output: 443.706

Approach: The given problem can be solved based on the following observations:

• A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. • In the above image above r is the radius of the circumcircle and A, B, C, and D are the lengths of the sides PQ, QR, RS, and SP respectively.
• The area of the quadrilateral is given by Bretschneider’s formula is: where, A, B, C, and D are the sides of the triangle and
α and γ are the opposite angles of the quadrilateral.

Since, the sum of opposite angles of the quadrilateral is 180 degree. Therefore, the value of cos(180/2) = cos(90) = 0.

Therefore, the formula for finding the area reduces to .

Therefore, the idea is to print the value of as the resultant area of the given quadrilateral.

Below is the implementation of the above approach:

## C++

 // C++ program for the above approach #include using namespace std; // Function to find the area// of cyclic quadrilateralfloat calculateArea(float A, float B,                    float C, float D){    // Stores the value of    // half of the perimeter    float S = (A + B + C + D) / 2;     // Stores area of cyclic quadrilateral    float area = sqrt((S - A) * (S - B)                      * (S - C) * (S - D));     // Return the resultant area    return area;} // Driver Codeint main(){    float A = 10;    float B = 15;    float C = 20;    float D = 25;    cout << calculateArea(A, B, C, D);     return 0;}

## Java

 // Java program for the above approachimport java.io.*; class GFG{     // Function to find the area// of cyclic quadrilateralstatic float calculateArea(float A, float B,                           float C, float D){         // Stores the value of    // half of the perimeter    float S = (A + B + C + D) / 2;     // Stores area of cyclic quadrilateral    float area = (float)Math.sqrt((S - A) * (S - B) *                                  (S - C) * (S - D));     // Return the resultant area    return area;} // Driver codepublic static void main (String[] args){    float A = 10;    float B = 15;    float C = 20;    float D = 25;         System.out.println(calculateArea(A, B, C, D)); }} // This code is contributed by Ankita saini

## Python3

 # Python3 program for the above approachfrom math import sqrt # Function to find the area# of cyclic quadrilateraldef calculateArea(A, B, C, D):         # Stores the value of    # half of the perimeter    S = (A + B + C + D) // 2     # Stores area of cyclic quadrilateral    area = sqrt((S - A) * (S - B) *                (S - C) * (S - D))     # Return the resultant area    return area # Driver Codeif __name__ == '__main__':         A = 10    B = 15    C = 20    D = 25         print(round(calculateArea(A, B, C, D), 3)) # This code is contributed by mohit kumar 29

## C#

 // C# program for the above approachusing System; class GFG{ // Function to find the area// of cyclic quadrilateralstatic float calculateArea(float A, float B,                           float C, float D){         // Stores the value of    // half of the perimeter    float S = (A + B + C + D) / 2;     // Stores area of cyclic quadrilateral    float area = (float)Math.Sqrt((S - A) * (S - B) *                                  (S - C) * (S - D));     // Return the resultant area    return area;} // Driver Codestatic public void Main(){    float A = 10;    float B = 15;    float C = 20;    float D = 25;         Console.Write(calculateArea(A, B, C, D));}} // This code is contributed by code_hunt

## Javascript

 
Output:
273.861

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

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