Find all angles of a given triangle - GeeksforGeeks

17 Feb, 2022
Given coordinates of all three vertices of the triangle in the 2D plane, the task is to find all three angles.Example:  
Input : A = (0, 0), 
        B = (0, 1), 
        C = (1, 0)
Output : 90, 45, 45
 
To solve this problem we use below Law of cosines.  
 
c^2 = a^2 + b^2 - 2(a)(b)(cos beta)
After re-arranging  
beta = acos( ( a^2 + b^2 - c^2 ) / (2ab) )
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. 
First, calculate the length of all the sides. 
Then apply above formula to get all angles in 
radian. Then convert angles from radian into 
degrees.
Below is implementation of above steps.  
C++
Java
Python3
C#
// Code to find all three angles// of a triangle given coordinate// of all three vertices#include <iostream>#include <utility> // for pair#include <cmath> // for math functionsusing namespace std; #define PI 3.1415926535 // returns square of distance b/w two pointsint lengthSquare(pair<int,int> X, pair<int,int> Y){    int xDiff = X.first - Y.first;    int yDiff = X.second - Y.second;    return xDiff*xDiff + yDiff*yDiff;} void printAngle(pair<int,int> A, pair<int,int> B,                pair<int,int> C){    // Square of lengths be a2, b2, c2    int a2 = lengthSquare(B,C);    int b2 = lengthSquare(A,C);    int c2 = lengthSquare(A,B);     // length of sides be a, b, c    float a = sqrt(a2);    float b = sqrt(b2);    float c = sqrt(c2);     // From Cosine law    float alpha = acos((b2 + c2 - a2)/(2*b*c));    float beta = acos((a2 + c2 - b2)/(2*a*c));    float gamma = acos((a2 + b2 - c2)/(2*a*b));     // Converting to degree    alpha = alpha * 180 / PI;    beta = beta * 180 / PI;    gamma = gamma * 180 / PI;     // printing all the angles    cout << "alpha : " << alpha << endl;    cout << "beta : " << beta << endl;    cout << "gamma : " << gamma << endl;} // Driver codeint main(){    pair<int,int> A = make_pair(0,0);    pair<int,int> B = make_pair(0,1);    pair<int,int> C = make_pair(1,0);     printAngle(A,B,C);     return 0;}
// Java Code to find all three angles// of a triangle given coordinate// of all three vertices import java.awt.Point;import static java.lang.Math.PI;import static java.lang.Math.sqrt;import static java.lang.Math.acos; class Test{    // returns square of distance b/w two points    static int lengthSquare(Point p1, Point p2)    {        int xDiff = p1.x- p2.x;        int yDiff = p1.y- p2.y;        return xDiff*xDiff + yDiff*yDiff;    }         static void printAngle(Point A, Point B,            Point C)    {    // Square of lengths be a2, b2, c2    int a2 = lengthSquare(B,C);    int b2 = lengthSquare(A,C);    int c2 = lengthSquare(A,B);         // length of sides be a, b, c    float a = (float)sqrt(a2);    float b = (float)sqrt(b2);    float c = (float)sqrt(c2);         // From Cosine law    float alpha = (float) acos((b2 + c2 - a2)/(2*b*c));    float betta = (float) acos((a2 + c2 - b2)/(2*a*c));    float gamma = (float) acos((a2 + b2 - c2)/(2*a*b));         // Converting to degree    alpha = (float) (alpha * 180 / PI);    betta = (float) (betta * 180 / PI);    gamma = (float) (gamma * 180 / PI);         // printing all the angles    System.out.println("alpha : " + alpha);    System.out.println("betta : " + betta);    System.out.println("gamma : " + gamma);    }         // Driver method    public static void main(String[] args)    {        Point A = new Point(0,0);        Point B = new Point(0,1);        Point C = new Point(1,0);              printAngle(A,B,C);    }}
# Python3 code to find all three angles# of a triangle given coordinate# of all three verticesimport math # returns square of distance b/w two pointsdef lengthSquare(X, Y):    xDiff = X[0] - Y[0]    yDiff = X[1] - Y[1]    return xDiff * xDiff + yDiff * yDiff     def printAngle(A, B, C):         # Square of lengths be a2, b2, c2    a2 = lengthSquare(B, C)    b2 = lengthSquare(A, C)    c2 = lengthSquare(A, B)     # length of sides be a, b, c    a = math.sqrt(a2);    b = math.sqrt(b2);    c = math.sqrt(c2);     # From Cosine law    alpha = math.acos((b2 + c2 - a2) /                         (2 * b * c));    betta = math.acos((a2 + c2 - b2) /                         (2 * a * c));    gamma = math.acos((a2 + b2 - c2) /                         (2 * a * b));     # Converting to degree    alpha = alpha * 180 / math.pi;    betta = betta * 180 / math.pi;    gamma = gamma * 180 / math.pi;     # printing all the angles    print("alpha : %f" %(alpha))    print("betta : %f" %(betta))    print("gamma : %f" %(gamma))         # Driver codeA = (0, 0)B = (0, 1)C = (1, 0) printAngle(A, B, C); # This code is contributed# by ApurvaRaj
// C# Code to find all three angles// of a triangle given coordinate// of all three verticesusing System;     class GFG{    class Point    {        public int x, y;        public Point(int x, int y)        {            this.x = x;            this.y = y;        }    }         // returns square of distance b/w two points    static int lengthSquare(Point p1, Point p2)    {        int xDiff = p1.x - p2.x;        int yDiff = p1.y - p2.y;        return xDiff * xDiff + yDiff * yDiff;    }         static void printAngle(Point A, Point B, Point C)    {        // Square of lengths be a2, b2, c2        int a2 = lengthSquare(B, C);        int b2 = lengthSquare(A, C);        int c2 = lengthSquare(A, B);                 // length of sides be a, b, c        float a = (float)Math.Sqrt(a2);        float b = (float)Math.Sqrt(b2);        float c = (float)Math.Sqrt(c2);                 // From Cosine law        float alpha = (float) Math.Acos((b2 + c2 - a2) /                                           (2 * b * c));        float betta = (float) Math.Acos((a2 + c2 - b2) /                                           (2 * a * c));        float gamma = (float) Math.Acos((a2 + b2 - c2) /                                           (2 * a * b));                 // Converting to degree        alpha = (float) (alpha * 180 / Math.PI);        betta = (float) (betta * 180 / Math.PI);        gamma = (float) (gamma * 180 / Math.PI);                 // printing all the angles        Console.WriteLine("alpha : " + alpha);        Console.WriteLine("betta : " + betta);        Console.WriteLine("gamma : " + gamma);    }         // Driver Code    public static void Main(String[] args)    {        Point A = new Point(0, 0);        Point B = new Point(0, 1);        Point C = new Point(1, 0);             printAngle(A, B, C);    }} // This code is contributed by Rajput-Ji
Output:  
alpha : 90
beta : 45
gamma : 45
Reference : https://en.wikipedia.org/wiki/Law_of_cosinesThis article is contributed by Pratik Chhajer . If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
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