# How to calculate the volume of tetrahedron given by 4 points

I want to calculate the volume of the tetrahedron defined by those 4 points:

$$P_1 = (-0.0865403, -0.122347, 0.898904)\\ P_2 = (-0.436523, -0.30131, 1.92251)\\ P_3 = (-0.459102, -0.0670386, 1.68168)\\ P_4 = (0,0,0)$$

How would you calculate this volume?

I'm doing

$$B_1 = P_1-P_4 = P_1\\B_2 = P_2-P_4 = P_2\\B_3 = P_3-P_4 = P_3$$

which is an equation that I found (is it correct?)

then

$$V = \frac{|B_1\cdot(B_2\times B_3)|}{6} = \frac{|(-0.0865403, -0.122347, 0.898904)\cdot((-0.436523, -0.30131, 1.92251)\times (-0.459102, -0.0670386, 1.68168))|}{6}$$

Which is 0.00786195 according to wolfram alpha (see here and divide by 6).

I have 2 questions: am I calculating the tetrahedron volume correctly? Does the order of points matter in the equation?

• The absolute value is invariant to the order of points. So for this case it it Ok Apr 9, 2020 at 5:05

A more convenient formula for the volume is $$V = \frac{1}{3!} \left|\begin{matrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \\ \end{matrix}\right|$$ where the vertices are $$(x_i, y_i, z_i)$$ for $$i = 1, 2, 3, 4$$ in any order. Take the absolute value of the result.
Abs[Det[{{-0.0865403, -0.122347, 0.898904, 1},

The advantage of this approach is that since one of your vertices is the origin, you could even compute the determinant fairly easily by hand as it reduces to a $$3 \times 3$$ determinant.