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 <string.h>
```
```
#include <stdio.h>
```
```
#include <stdlib.h>
```
```
 
```
```
int i, j, c;
```
```
double det(int n, double mat[3][3]) {
```
```
    double submat[3][3];
```
```
    float d;
```
```
    for (c = 0; c < n; c++) {
```

        int subi = 0; //submatrix's i value

```
        for (i = 1; i < n; i++) {
```
```
            int subj = 0;
```
```
            for (j = 0; j < n; j++) {
```
```
                if (j == c)
```
```
                    continue;
```

                submat[subi][subj] = mat[i][j];

```
                subj++;
```
```
            }
```
```
            subi++;
```
```
 
```
```
        }
```

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

```
    }
```
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    return d;
```
```
}
```
```
 
```
```
int main(int argc, char **argv) {
```
```
 
```

    printf("Enter the points of the triangle:\n");

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

```
    scanf("%d", &x1);
```
```
    scanf("%d", &y1);
```
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    scanf("%d", &z1);
```
```
    scanf("%d", &x2);
```
```
    scanf("%d", &y2);
```
```
    scanf("%d", &z2);
```
```
    scanf("%d", &x3);
```
```
    scanf("%d", &y3);
```
```
    scanf("%d", &z3);
```
```
    scanf("%d", &x4);
```
```
    scanf("%d", &y4);
```
```
    scanf("%d", &z4);
```
```
    double mat[4][4];
```
```
    mat[0][0] = x1;
```
```
    mat[0][1] = x2;
```
```
    mat[0][2] = x3;
```
```
    mat[0][3] = x4;
```
```
    mat[1][0] = y1;
```
```
    mat[1][1] = y2;
```
```
    mat[1][2] = y3;
```
```
    mat[1][3] = y4;
```
```
    mat[2][0] = z1;
```
```
    mat[2][1] = z2;
```
```
    mat[2][2] = z3;
```
```
    mat[2][3] = z4;
```
```
    mat[3][0] = 1;
```
```
    mat[3][1] = 1;
```
```
    mat[3][2] = 1;
```
```
    mat[3][3] = 1;
```
```
 
```

    printf("\nMatrix formed by the points: \n");

```
    for (i = 0; i < 4; i++) {
```
```
        for (j = 0; j < 4; j++) {
```
```
            printf("%lf ", mat[i][j]);
```
```
        }
```
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        printf("\n");
```
```
    }
```
```
    double matrix[3][3];
```
```
 
```
```
    matrix[0][0] = x1 - x4;
```
```
    matrix[0][1] = x2 - x4;
```
```
    matrix[0][2] = x3 - x4;
```
```
    matrix[1][0] = y1 - y4;
```
```
    matrix[1][1] = y2 - y4;
```
```
    matrix[1][2] = y3 - y4;
```
```
    matrix[2][0] = z1 - z4;
```
```
    matrix[2][1] = z2 - z4;
```
```
    matrix[2][2] = z3 - z4;
```
```
    for (i = 0; i < 3; i++) {
```
```
        for (j = 0; j < 3; j++) {
```
```
            printf("%lf ", mat[i][j]);
```
```
        }
```
```
        printf("\n");
```
```
    }
```

    float determinant = det(3, matrix) / 6;

```
    if (determinant < 0)
```

        printf("The area of tetrahedron formed by (%d, %d, %d), (%d, %d, %d), (%d, %d, %d), (%d, %d, %d) = %lf ",

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

```
                (determinant * -1));
```
```
 
```
```
    else
```

        printf("The area of tetrahedron formed by (%d, %d, %d), (%d, %d, %d), (%d, %d, %d), (%d, %d, %d) = %lf ",

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

```
    return 0;
```
```
}
```

Output:

$ gcc TetrahedronVolume.c
$ ./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

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