1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Triple Integral - Volume of Tetrahedron

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Actually, the problem was addressed in a prior post:


    Which is closed.

    2. Relevant equations

    I would like to know how HallsofIvy (or anyone) arrived at the formula for the tetrahedron given the vertices (1,0,0), (0,2,0), (0,0,3).

    Ultimately I am to find the volume of this tetrahedron using triple integrals.

    But I'm not worried about the integral as much as the setup:

    The equation I get is 3 -3x -3/2y
    not 1 -3x -3/2y

    3. The attempt at a solution
    I've taken two vectors from these points, taken their cross product, and created an equation of a plane. I am still getting my answer, and consequently an integral that doesn't seem right!

    -Dave K
  2. jcsd
  3. Apr 30, 2012 #2


    User Avatar
    Gold Member

    You should label each of the vertices. Let A=(1,0,0), B=(0,2,0), C=(0,0,3). There are 2 methods (that i know of):

    First method:
    The Cartesian equation of plane is: [itex]ax +by+cz=d[/itex]

    Just plug in the coordinates of the 3 points and solve the system of 3 linear equations. You should get x, y and z in terms of d. Then, divide throughout by d to get the final equation of the plane.

    Second method (what you're expected to use):

    Find two vectors that lie in the plane. Do the cross product to get the normal vector, [itex]\vec n[/itex].

    Then, use the formula: [itex]\vec r.\vec n=\vec a.\vec n[/itex] where [itex]\vec a[/itex] is any point found in the plane.

    Using the second method, you will get the Cartesian equation of the plane: [itex]6x+3y+2z=6[/itex]

    There is indeed a mistake in that post: https://www.physicsforums.com/showpost.php?p=1387411&postcount=3

    Last edited: Apr 30, 2012
  4. Apr 30, 2012 #3
    Oh thank goodness.

    I wasn't trusting my own answer, and it (the correct answer) makes the integration a little bit uglier.

    Thanks man.


    -Dave K
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook