Results 1 to 5 of 5

Math Help - Tetrahedron volume

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    296

    Tetrahedron volume

    Consider the tetrahedron with vertices
    (0,0,0)
    (0,1,2)
    (0,2,0)
    (3,0,0)

    Find its volume using integration. (No double or triple integrals though)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    Find a formula in terms of z for the area of a cross-section of the tetrahedron by a plane parallel to the xy-plane, and then integrate:

    \int _0 ^2 A(z) dz,

    where A(z) is the area of the cross-section.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2009
    Posts
    296
    Thanks, but I'm not sure how to find A(z)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    First, observe that the dimensions of the triangle decrease linearly as you move from z = 0 to z = 2. Therefore,

    The length in x is equal to 3 - 1.5z

    The length in y is equal to 2 - z

    The area of the triangular cross-section is 0.5(3 - 1.5z)(2 - z).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jan 2009
    Posts
    296
    Okay, I see what you did. Sorry, but the only thing I still don't understand is how you got 3-1.5z. I understand the 2-z I think, but not this part.

    EDIT::Nevermind, got it!
    Last edited by zhupolongjoe; February 10th 2010 at 06:39 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Calculate the volume of the tetrahedron
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 27th 2011, 01:51 AM
  2. Volume of a tetrahedron
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 7th 2010, 04:38 AM
  3. Tetrahedron volume
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 16th 2009, 12:45 PM
  4. Volume of a tetrahedron
    Posted in the Geometry Forum
    Replies: 1
    Last Post: April 9th 2008, 05:06 AM
  5. Volume of a tetrahedron
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 27th 2008, 11:50 PM

Search Tags


/mathhelpforum @mathhelpforum