Find interior angles for each side of a given Cyclic Quadrilateral

Given four positive integers A, B, C, and D representing the sides of a Cyclic Quadrilateral, the task is to find all the interior angles of the cyclic quadrilateral.

A cyclic quadrilateral is a quadrilateral whose vertices lie on a single circle. 
This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic(A, B, C, and D). 
( In the figure, r is the circumradius and a, b, c, and d are length of AB, BC, CD, and DA respectively).

Examples:

Input: A = 10, B = 15, C = 20, D = 25
Output:
?A: 85.59 degrees
?B: 122.58 degrees
?C: 94.41 degrees
?D: 57.42 degrees

Input: A = 10, B = 10, C = 10, D = 10
Output:
?A: 90.00 degrees
?B: 90.00 degrees
?C: 90.00 degrees
?D: 90.00 degrees

Approach: The given problem can be solved by using the formula to calculate the cosine of the interior angle of a cyclic quadrilateral. The formula is given by:

Follow the steps below to solve the problem:

• Store the cosine of each interior angle of the cyclic quadrilateral.
• Find the angle in radians using acos() function.
• Convert the angle in radians to degrees and print the result.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the interior angles
// of the cyclic quadrilateral
void findAngles(double a, double b,
                double c, double d)
{
    // Stores the numerator and the
    // denominator to find angle A
    double numerator = a * a + d * d
                       - b * b - c * c;

    double denominator = 2 * (a * b + c * d);

    double x = numerator / denominator;

    cout << fixed << setprecision(2)
         << "A: " << (acos(x) * 180) / 3.141592
         << " degrees";

    // Stores the numerator and the
    // denominator to find angle B
    numerator = a * a + b * b
                - c * c - d * d;

    x = numerator / denominator;

    cout << fixed << setprecision(2)
         << "\nB: " << (acos(x) * 180) / 3.141592
         << " degrees";

    // Stores the numerator and the
    // denominator to find angle C:
    numerator = c * c + b * b
                - a * a - d * d;

    x = numerator / denominator;

    cout << fixed << setprecision(2)
         << "\nC: " << (acos(x) * 180) / 3.141592
         << " degrees";

    // Stores the numerator and the
    // denominator to find angle D:
    numerator = d * d + c * c
                - a * a - b * b;

    x = numerator / denominator;

    cout << fixed << setprecision(2)
         << "\nD: " << (acos(x) * 180) / 3.141592
         << " degrees";
}

// Driver Code
int main()
{
    double A = 10, B = 15, C = 20, D = 25;
    findAngles(A, B, C, D);

    return 0;
}

Java

// Java program for the above approach
class GFG{

// Function to find the interior angles
// of the cyclic quadrilateral
static void findAngles(double a, double b,
                       double c, double d)
{

    // Stores the numerator and the
    // denominator to find angle A
    double numerator = a * a + d * d -
                       b * b - c * c;

    double denominator = 2 * (a * b + c * d);

    double x = numerator / denominator;

    System.out.println("A: " +
       Math.round(((Math.acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle B
    numerator = a * a + b * b - c * c - d * d;

    x = numerator / denominator;

    System.out.println("B: " +
       Math.round(((Math.acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle C:
    numerator = c * c + b * b -
                a * a - d * d;

    x = numerator / denominator;

    System.out.println("C: " +
       Math.round(((Math.acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle D:
    numerator = d * d + c * c -
                a * a - b * b;

    x = numerator / denominator;

    System.out.println("D: " +
       Math.round(((Math.acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");
}

// Driver Code
public static void main (String[] args)
{
    double A = 10, B = 15, C = 20, D = 25;

    findAngles(A, B, C, D);
}
}

// This code is contributed by AnkThon

Python3

# Python3 program for the above approach
import math

# Function to find the interior angles
# of the cyclic quadrilateral
def findAngles(a, b, c, d):

    # Stores the numerator and the
    # denominator to find angle A
    numerator = a * a + d * d - b * b - c * c
    denominator = 2 * (a * b + c * d)
    x = numerator / denominator
    print("A: ", '%.2f' % ((math.acos(x) * 180) /
          3.141592), " degrees")

    # Stores the numerator and the
    # denominator to find angle B
    numerator = a * a + b * b - c * c - d * d
    x = numerator / denominator
    print("B: ", '%.2f' % ((math.acos(x) * 180) /
          3.141592), " degrees")

    # Stores the numerator and the
    # denominator to find angle C:
    numerator = c * c + b * b - a * a - d * d
    x = numerator / denominator
    print("C: ", '%.2f' % ((math.acos(x) * 180) /
          3.141592), " degrees")

    # Stores the numerator and the
    # denominator to find angle D:
    numerator = d * d + c * c - a * a - b * b
    x = numerator / denominator
    print("D: ", '%.2f' % ((math.acos(x) * 180) /
          3.141592), " degrees")

# Driver Code
if __name__ == "__main__":

    A = 10
    B = 15
    C = 20
    D = 25

    findAngles(A, B, C, D)

# This code is contributed by ukasp

C#

// C# program for the above approach
using System;

class GFG{

// Function to find the interior angles
// of the cyclic quadrilateral
static void findAngles(double a, double b,
                       double c, double d)
{

    // Stores the numerator and the
    // denominator to find angle A
    double numerator = a * a + d * d -
                       b * b - c * c;

    double denominator = 2 * (a * b + c * d);

    double x = numerator / denominator;

    Console.WriteLine("A: " +
       Math.Round(((Math.Acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle B
    numerator = a * a + b * b - c * c - d * d;

    x = numerator / denominator;

    Console.WriteLine("B: " +
       Math.Round(((Math.Acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle C:
    numerator = c * c + b * b -
                a * a - d * d;

    x = numerator / denominator;

    Console.WriteLine("C: " +
       Math.Round(((Math.Acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");

    // Stores the numerator and the
    // denominator to find angle D:
    numerator = d * d + c * c -
                a * a - b * b;

    x = numerator / denominator;

    Console.WriteLine("D: " +
       Math.Round(((Math.Acos(x) * 180) /
                       3.141592) * 100.0) /
                       100.0 + " degrees");
}

// Driver Code
public static void Main(string[] args)
{
    double A = 10, B = 15, C = 20, D = 25;

    findAngles(A, B, C, D);
}
}

// This code is contributed by AnkThon

Javascript

<script>

// JavaScript program for the above approach

// Function to find the interior angles
// of the cyclic quadrilateral
function findAngles(a, b, c, d){

    // Stores the numerator and the
    // denominator to find angle A
    var numerator = a * a + d * d - b * b - c * c
    var denominator = 2 * (a * b + c * d)
    var x = numerator / denominator
    document.write("A: ", Math.round(((Math.acos(x) * 180) /
                       3.141592) * 100) / 100.0, " degrees");
    document.write("<br>");

    // Stores the numerator and the
    // denominator to find angle B
    numerator = a * a + b * b - c * c - d * d
    x = numerator / denominator
    document.write("B: ",  Math.round(((Math.acos(x) * 180) /
          3.141592) * 100) / 100.0, " degrees");
    document.write("<br>");

    // Stores the numerator and the
    // denominator to find angle C:
    numerator = c * c + b * b - a * a - d * d
    x = numerator / denominator
    document.write("C: ", Math.round(((Math.acos(x) * 180) /
          3.141592)  * 100) / 100.0, " degrees");
    document.write("<br>");

    // Stores the numerator and the
    // denominator to find angle D:
    numerator = d * d + c * c - a * a - b * b
    x = numerator / denominator
    document.write("D: ", Math.round(((Math.acos(x) * 180) /
          3.141592)  * 100) / 100.0, " degrees");
}

// Driver Code
var A = 10
var B = 15
var C = 20
var D = 25

findAngles(A, B, C, D)

// This code is contributed by AnkThon

</script>

Output:

A: 85.59 degrees
B: 122.58 degrees
C: 94.41 degrees
D: 57.42 degrees

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