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Volume of tetrahedron when you are given four planes

  1. Nov 28, 2007 #1
    1. The problem statement, all variables and given/known data
    I have to find volume of tetrahedron that is bounded between 4 planes.
    Planes are

    2. Relevant equations
    3. The attempt at a solution

    I found four points where planes meet. These are:

    From that I made vectors AB, AC, AD and then I put that into [tex]\vec{a}[/tex][tex]\ast[/tex]([tex]\vec{b}[/tex][tex]\times[/tex][tex]\vec{c}[/tex]) and got that volume of parallelepiped is 4. From there I got that volume of this tetrahedron is 2/3. Is this the correct and shortest way to get a solution? My teacher said that I can use formula V=B*v/3 but I don't know where to use it.
    Last edited: Nov 29, 2007
  2. jcsd
  3. Nov 28, 2007 #2


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    I can't tell you where you would use B= B*v/2 since you haven't said what B or v mean in that formula!
  4. Nov 29, 2007 #3
    Sorry. That's Croatian notation. I think american would be Volume=1/3*B*h where B is area of the base and h is height of tetrahedron. I can calculate h from formula for distance between point where first three planes intersect and the fourth plane. I don't know how to calculate area of the base. Is it correct that volume of this tetrahedron is 2/3?
    Last edited: Nov 29, 2007
  5. Nov 30, 2007 #4


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    You can choose any 3 of the 4 vertices to be a triangular base. A quick way of finding the area is to construct vectors [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] from one of the vertices to the other two. Then the area of the base, the triangle, is [itex]B= (1/2)|\vec{u}\times\vec{v}|[/itex]. The height of the distance from the fourth point to the plane defined by the first three points.
    Last edited: Sep 22, 2011
  6. Sep 22, 2011 #5

    You can find out the volume by this formula but it is difficult to calculate the determinant of a 4*4 matrix
  7. Sep 22, 2011 #6


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    @borovcm Don't put tex tags around every expression. Type whatever equations would nicely fit on one line and put the tags around that.
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