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Thread: Calculate volume of irregular triangular pyramid using differences between prisms

  1. #1
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    Question Calculate volume of irregular triangular pyramid using differences between prisms

    Here's a picture to better depict what I'm talking about:

    Calculate volume of irregular triangular pyramid using differences between prisms-volume.png

    The red shape is the irregular triangular pyramid, the blue shape is the triangular prism.

    All vertices of the pyramid have different z-values (except the ones shared with the prism).

    I could of course calculate the volume of the pyramid and the prism independently and add them together, but I'm looking for a simpler solution that involves a "single" operation (this is for thousands of iterations of data with poorly distinguished vertices between batches; two independent volume calculations is out of the question).

    Could I calculate the volume of two prisms -- one with a height at the bottom of the pyramid, one with the height at its top -- take the difference, and then operate on this "slice" of prism using the slope of a face of the pyramid to calculate the volume of that pyramid? If so, what would my formula look like (using the x, y, and z values of the pyramid's vertices)? Or could there be some other way?

    Thanks!
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  2. #2
    MHF Contributor
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    Re: Calculate volume of irregular triangular pyramid using differences between prisms

    This is not a trivial problem. The methods that I would try would likely use multivariable calculus. There may be a geometric approach, but it is likely difficult to find and may not be apparent without first an investigation through calculus.
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