Calculate Volume of Tetrahedron with Given Vertices | Step-by-Step Solution

[madachi]

Homework Statement

Find the volume of the tetrahedron with vertices at $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$

The Attempt at a Solution

I worked out the triple integral and found out that the volume is $\frac{1}{6}$? Is this correct? I know there is probably a much quicker way working the volume by just using the volume formula for tetrahedron. However, I am not sure which value to substitute to the formula, so could you just tell me whether this answer is right or not?

Thanks!

 

[Mr.Miyagi]

The vertices you give do not make a tetrahedron. Try to draw the points in three dimensions. You'll see the volume is just half that of a cube with length 1. So the volume you're seeking should be 1/2.

 

[Dick]

For a tetrahedron, like a cone, the area is (1/3)*(area of the base)*height. So (1/3)*(1/2)*1=1/6, yes.

 

[HallsofIvy]

Mr. Miyagi is wrong. That is in fact a tetrahedron, it is 1/6 of a cube, not 1/2, and its volume is, indeed, 1/6.

More generally, the volume of the tetrahedron is with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c) is (abc)/6.

 

[Mr.Miyagi]

Ugh, sorry about that... Is it too late to claim temporary insanity? :uhh:

Thanks for correcting it so quickly.