Question

Find the volume of the tetrahedron with the given vertices. (2, 0, 0), (0, 4, 0), (0, 0, 4), (2, 4, 4)

Answer 1

Answer:

The volume of the tetrahedron is

$\frac{-32}{3}$

Step-by-step explanation:

To find the volume of a tetrahedron with vertices (2, 0, 0), (0, 4, 0), (0, 0, 4), (2, 4, 4).

We know by definition that the volume of a tetrahedron with vertices $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3})$ is

$\frac{1}{6}\begin{vmatrix}2 & 0 & 0 & 1\\0 & 4 & 0 & 1\\0 & 0 & 4 & 1\\2 & 4 & 4 & 1\end{vmatrix}$

From here we have:

$\frac{1}{6}\begin{vmatrix}x_1 & y_1 & z_1 & 1\\x_2 & y_2 & y_2 & 1\\x_3 & y_3 & z_3 & 1\\x_4 & y_4 & y_4 & 1\end{vmatrix}$

Finding the determinant of that matrix we have:

$\frac{1}{6} (-64)$ = $\frac{-32}{3}$