Find all angles of a triangle in 3D

Given coordinates of 3 vertices of a triangle in 3D i.e. A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3). The task is to find out all the angles of the triangle formed by above coordinates.

Examples:

Input: 
x1 = -1, y1 = 3, z1 = 2
x2 = 2, y2 = 3, z2 = 5
x3 = 3, y3 = 5, z3 = -2

Output:
angle A =  90.0 degree
angle B =  54.736 degree
angle C =  35.264 degree

Approach:
For finding angle A find out direction ratios of AB and AC :
direction ratios of AB = x2-x1, y2-y1, z2-z1
direction ratios of AC = x3-x1, y3-y1, z3-z1
then angle A = cos^{-1}\left(\frac{(x_{2}-x_{1})*(x_{3}-x_{1})+(y_{2}-y_{1})*(y_{3}-y_{1})+(z_{2}-z_{1})*(z_{3}-z_{1})}{(\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}})*(\sqrt{(x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2}+(z_{3}-z_{1})^{2}})}\right)

For finding angle B find out direction ratios of BA and BC :
direction ratios of BA = x1-x2, y1-y2, z1-z2
direction ratios of BC = x3-x2, y3-y2, z3-z2
then angle B = cos^{-1}\left(\frac{(x_{1}-x_{2})*(x_{3}-x_{2})+(y_{1}-y_{2})*(y_{3}-y_{2})+(z_{1}-z_{2})*(z_{3}-z_{2})}{(\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}})*(\sqrt{(x_{3}-x_{2})^{2}+(y_{3}-y_{2})^{2}+(z_{3}-z_{2})^{2}})}\right)



For finding angle C find out direction ratios of CB and CA :
direction ratios of CB = x2-x3, y2-y3, z2-z3
direction ratios of CA = x1-x3, y1-y3, z1-z3
then angle C = cos^{-1}\left(\frac{(x_{2}-x_{3})*(x_{1}-x_{3})+(y_{2}-y_{3})*(y_{1}-y_{3})+(z_{2}-z_{3})*(z_{1}-z_{3})}{(\sqrt{(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}+(z_{2}-z_{3})^{2}})*(\sqrt{(x_{1}-x_{3})^{2}+(y_{1}-y_{3})^{2}+(z_{1}-z_{3})^{2}})}\right)

Below is the implementation of above approach:

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

//CPP program for finding all angles of a triangle 
#include<bits/stdc++.h>
#include<cmath>
using namespace std;
  
// function for finding the angle 
float angle_triangle(int x1, int x2, int x3, 
            int y1, int y2, int y3, int z1, int z2, int z3)
{
  
    int num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) ;
  
    float den = sqrt(pow((x2-x1),2)+pow((y2-y1),2)+pow((z2-z1),2))*\
                sqrt(pow((x3-x1),2)+pow((y3-y1),2)+pow((z3-z1),2)) ;
  
    float angle = acos(num / den)*(180.0/3.141592653589793238463) ;
  
    return angle ;
}
  
// Driver code
int main()
{
int x1 = -1;
int y1 = 3;
int z1 = 2;
int x2 = 2;
int y2 = 3;
int z2 = 5;
int x3 = 3;
int y3 = 5;
int z3 = -2;
float angle_A = angle_triangle(x1, x2, x3, y1, y2,
                                y3, z1, z2, z3);
float angle_B = angle_triangle(x2, x3, x1, y2, y3, 
                                y1, z2, z3, z1);
float angle_C = angle_triangle(x3, x2, x1, y3, 
                            y2, y1, z3, z2, z1);
cout<<"Angles are :"<<endl; 
cout<<setprecision(3);
cout<<"angle A = "<<angle_A<<" degree"<<endl;
cout<<"angle B = "<<angle_B<<" degree"<<endl;
cout<<"angle C = "<<angle_C<< " degree"<<endl;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

//Java program for finding all angles of a triangle 
  
class GFG{
// function for finding the angle 
static double angle_triangle(int x1, int x2, int x3, 
            int y1, int y2, int y3, int z1, int z2, int z3)
{
  
    int num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) ;
  
    double den = Math.sqrt(Math.pow((x2-x1),2)+
                Math.pow((y2-y1),2)+Math.pow((z2-z1),2))*
                Math.sqrt(Math.pow((x3-x1),2)+
                Math.pow((y3-y1),2)+Math.pow((z3-z1),2)) ;
  
    double angle = Math.acos(num / den)*(180.0/3.141592653589793238463) ;
  
    return angle ;
}
  
// Driver code
public static void main(String[] args)
{
int x1 = -1;
int y1 = 3;
int z1 = 2;
int x2 = 2;
int y2 = 3;
int z2 = 5;
int x3 = 3;
int y3 = 5;
int z3 = -2;
double angle_A = angle_triangle(x1, x2, x3, y1,
                            y2, y3, z1, z2, z3);
double angle_B = angle_triangle(x2, x3, x1,
                            y2, y3, y1, z2, z3, z1);
double angle_C = angle_triangle(x3, x2, x1,
                            y3, y2, y1, z3, z2, z1);
System.out.println("Angles are :"); 
System.out.println("angle A = "+angle_A+" degree");
System.out.println("angle B = "+angle_B+" degree");
System.out.println("angle C = "+angle_C+" degree");
}
}
// This code is contributed by mits

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python Code for finding all angles of a triangle 
import math 
  
# function for finding the angle 
def angle_triangle(x1, x2, x3, y1, y2, y3, z1, z2, z3): 
  
    num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) 
  
    den = math.sqrt((x2-x1)**2+(y2-y1)**2+(z2-z1)**2)*\
                math.sqrt((x3-x1)**2+(y3-y1)**2+(z3-z1)**2
  
    angle = math.degrees(math.acos(num / den)) 
  
    return round(angle, 3
  
# driver code     
x1 = -1
y1 = 3
z1 = 2
x2 = 2
y2 = 3
z2 = 5
x3 = 3
y3 = 5
z3 = -2
angle_A = angle_triangle(x1, x2, x3, y1, y2, y3, z1, z2, z3) 
angle_B = angle_triangle(x2, x3, x1, y2, y3, y1, z2, z3, z1) 
angle_C = angle_triangle(x3, x2, x1, y3, y2, y1, z3, z2, z1) 
print("Angles are :"
print("angle A = ", angle_A, "degree"
print("angle B = ", angle_B, "degree"
print("angle C = ", angle_C, "degree"

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program for finding all 
// angles of a triangle 
using System;
                      
class GFG
{
      
// function for finding the angle 
static double angle_triangle(int x1, int x2, int x3, 
                             int y1, int y2, int y3, 
                             int z1, int z2, int z3)
{
  
    int num = (x2 - x1) * (x3 - x1) +
              (y2 - y1) * (y3 - y1) + 
              (z2 - z1) * (z3 - z1);
  
    double den = Math.Sqrt(Math.Pow((x2 - x1), 2) +
                           Math.Pow((y2 - y1), 2) + 
                           Math.Pow((z2 - z1), 2)) *
                 Math.Sqrt(Math.Pow((x3 - x1), 2) + 
                           Math.Pow((y3 - y1), 2) + 
                           Math.Pow((z3 - z1), 2));
  
    double angle = Math.Acos(num / den) * 
                   (180.0/3.141592653589793238463);
  
    return angle ;
}
  
// Driver code
public static void Main()
{
    int x1 = -1, y1 = 3, z1 = 2;
    int x2 = 2, y2 = 3, z2 = 5;
    int x3 = 3, y3 = 5, z3 = -2;
    double angle_A = angle_triangle(x1, x2, x3, 
                                    y1, y2, y3, 
                                    z1, z2, z3);
    double angle_B = angle_triangle(x2, x3, x1,
                                    y2, y3, y1, 
                                    z2, z3, z1);
    double angle_C = angle_triangle(x3, x2, x1,
                                    y3, y2, y1,
                                    z3, z2, z1);
    Console.WriteLine("Angles are :"); 
    Console.WriteLine("angle A = " + angle_A + 
                                   " degree");
    Console.WriteLine("angle B = " + angle_B +
                                   " degree");
    Console.WriteLine("angle C = " + angle_C + 
                                   " degree");
}
}
  
// This code is contributed by 29AjayKumar

chevron_right


PHP

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program for finding all 
// angles of a triangle 
  
// function for finding the angle 
function angle_triangle($x1, $x2, $x3
                        $y1, $y2, $y3
                        $z1, $z2, $z3)
{
  
    $num = ($x2 - $x1) * ($x3 - $x1) + 
           ($y2 - $y1) * ($y3 - $y1) + 
           ($z2 - $z1) * ($z3 - $z1);
  
    $den = sqrt(pow(($x2 - $x1), 2) + 
                pow(($y2 - $y1), 2) + 
                pow(($z2 - $z1), 2)) *
           sqrt(pow(($x3 - $x1), 2) + 
                pow(($y3 - $y1), 2) + 
                pow(($z3 - $z1), 2));
  
    $angle = acos($num / $den) * 
                 (180.0 / 3.141592653589793238463);
  
    return $angle ;
}
  
// Driver code
$x1 = -1; $y1 = 3; $z1 = 2;
$x2 = 2; $y2 = 3; $z2 = 5;
$x3 = 3; $y3 = 5; $z3 = -2;
$angle_A = angle_triangle($x1, $x2, $x3, $y1, $y2,
                          $y3, $z1, $z2, $z3);
$angle_B = angle_triangle($x2, $x3, $x1, $y2, $y3
                          $y1, $z2, $z3, $z1);
$angle_C = angle_triangle($x3, $x2, $x1, $y3
                          $y2, $y1, $z3, $z2, $z1);
echo "Angles are :\n"
echo "angle A = " . round($angle_A, 3) . " degree\n";
echo "angle B = " . round($angle_B, 3) . " degree\n";
echo "angle C = " . round($angle_C, 3) . " degree\n";
  
// This code is contributed by mits
?>

chevron_right


Output:

Angles are :
angle A =  90.0 degree
angle B =  54.736 degree
angle C =  35.264 degree

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.




My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.