C++ Program to Find the Volume of a Tetrahedron Using Determinants

This is a C++ Program to find the volume of tetrahedron.
Call the four vertices of the tetrahedron (a, b, c), (d, e, f), (g, h, i), and (p, q, r). Now create a 4-by-4 matrix in which the coordinate triples form the colums of the matrix, with a row of 1’s appended at the bottom:
a d g p
b e h q
c f i r
1 1 1 1
The volume of the tetrahedron is 1/6 times the absolute value of the matrix determinant. For any 4-by-4 matrix that has a row of 1’s along the bottom, you can compute the determinant with a simplification formula that reduces the problem to a 3-by-3 matrix
a-p d-p g-p
b-q e-q h-q
c-r f-r i-r

Here is source code of the C++ Program to Compute the Volume of a Tetrahedron Using Determinants. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below.

```
#include<stdio.h>
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#include<stdlib.h>
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#include<iostream>
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#include<math.h>
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using namespace std;
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double det(int n, double mat[3][3])
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{
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    double submat[3][3];
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    float d;
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    for (int c = 0; c < n; c++)
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    {
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        int subi = 0; //submatrix's i value

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        for (int i = 1; i < n; i++)
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        {
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            int subj = 0;
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            for (int j = 0; j < n; j++)
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            {
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                if (j == c)
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                    continue;
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                submat[subi][subj] = mat[i][j];

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                subj++;
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            }
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            subi++;
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        }
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        d = d + (pow(-1, c) * mat[0][c] * det(n - 1, submat));

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    }
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    return d;
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}
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int main(int argc, char **argv)
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{
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    cout << "Enter the points of the triangle:\n";

    int x1, x2, x3, x4, y1, y2, y3, y4, z1, z2, z3, z4;

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    cin >> x1;
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    cin >> x2;
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    cin >> x3;
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    cin >> x4;
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    cin >> y1;
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    cin >> y2;
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    cin >> y3;
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    cin >> y4;
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    cin >> z1;
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    cin >> z2;
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    cin >> z3;
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    cin >> z4;
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    double mat[4][4];
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    mat[0][0] = x1;
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    mat[0][1] = x2;
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    mat[0][2] = x3;
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    mat[0][3] = x4;
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    mat[1][0] = y1;
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    mat[1][1] = y2;
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    mat[1][2] = y3;
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    mat[1][3] = y4;
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    mat[2][0] = z1;
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    mat[2][1] = z2;
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    mat[2][2] = z3;
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    mat[2][3] = z4;
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    mat[3][0] = 1;
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    mat[3][1] = 1;
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    mat[3][2] = 1;
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    mat[3][3] = 1;
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    cout << "\nMatrix formed by the points: \n";

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    for (int i = 0; i < 4; i++)
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    {
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        for (int j = 0; j < 4; j++)
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        {
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            cout << mat[i][j] << " ";
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        }
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        cout << endl;
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    }
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    double matrix[3][3];
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    matrix[0][0] = x1 - x4;
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    matrix[0][1] = x2 - x4;
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    matrix[0][2] = x3 - x4;
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    matrix[1][0] = y1 - y4;
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    matrix[1][1] = y2 - y4;
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    matrix[1][2] = y3 - y4;
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    matrix[2][0] = z1 - z4;
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    matrix[2][1] = z2 - z4;
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    matrix[2][2] = z3 - z4;
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    for (int i = 0; i < 3; i++)
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    {
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        for (int j = 0; j < 3; j++)
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        {
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            cout << matrix[i][j] << " ";

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        }
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        cout << endl;
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    }
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    float determinant = det(3, matrix) / 6;

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    if (determinant < 0)
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        cout << "The Area of the tetrahedron formed by (" << x1 << "," << y1

                << "," << z1 << "), (" << x2 << "," << y2 << "," << z2

                << "), (" << x3 << "," << y3 << "," << z3 << "), (" << x4 << ","

                << y4 << "," << z4 << ") = " << (determinant * -1);

```
    else
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        cout << "The Area of the tetrahedron formed by (" << x1 << "," << y1

                << "," << z1 << "), (" << x2 << "," << y2 << "," << z2

                << "), (" << x3 << "," << y3 << "," << z3 << "), (" << x4 << ","

                << y4 << "," << z4 << ") = " << determinant;

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    return 0;
```
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}
```

Output:

$ g++ TetrahedronVolume.cpp
$ a.out
 
Enter the points of the triangle:
0 9 6 0
4 2 1 1
3 4 7 5
 
Matrix formed by the points: 
0 9 6 0 
4 2 1 1 
3 4 7 5 
1 1 1 1 
 
0 9 6 
3 1 0 
-2 -1 2 
The Area of the tetrahedron formed by (0,4,3), (9,2,4), (6,1,7), (0,1,5) = 10.0
------------------
(program exited with code: 0)
Press return to continue

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