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Volume of tetrahedron with 5 vertices

  1. Aug 3, 2015 #1
    I already found how to calculate the volume of tetrahedron from 4 vertices, i.e. V = 1/6(dot(d1,D), where D = cross(d2,d3).
    Could somebody specify the formula or an article for volume of tetrahedron using 5 vertices, A = (x1, y1, z1), B = (x2, y2, z2), C = (x3, y3, z3), D = (x4, y4, z4) and O = (x0,y0,z0) .
    Thank you very much in advance.
     
  2. jcsd
  3. Aug 3, 2015 #2

    jedishrfu

    Staff: Mentor

    Welcome to PF!

    A tetrahedron has exactly four vertices, when you add another you get a pentahedron. One such example is the pyramid.

    Another is when you put two tetrahedrons together at the common surface,
     
  4. Aug 3, 2015 #3
    Thank you Jedishrfu!!
    I checked for volume of pyramid, but could not find the ways to Calculate the Volume of "irregular pyramid".
    as, the base of an irregular pyramid is an irregular polygon, and as a result, its faces are not equally sized. any ideas regarding that?

    Or in case 2 as you mentioned, should I consider one point as common surface and calculate two separate volumes of two tetrahedron and add them?
     
  5. Aug 3, 2015 #4

    jedishrfu

    Staff: Mentor

    In the common surface example, yes, I think you'd calculate the volume for each one and then add them to get the volume of that particular pentahedron.
     
  6. Aug 3, 2015 #5
    Thanks!!
    it works, in case the fourth point is lying on the same line connecting two points. for example, mid point of the line connecting two points.
    In case the point is lying outside, then it becomes the irregular pyramid.

    Still searching the formula for Volume of the irregular pyramid.
     
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