Volume of Irregular Pyramid of Human ArmDate: 7/24/96 at 9:38:7 From: Anonymous Subject: Volume of Irregular Pyramid Formed by Body I have an irregular pyramid formed by moving my arm and forearm in space. Consider the arm and forearm as two sides of a triangle whose third side is the length between shoulder and wrist in the rest position (forearm rested on a table), moving this triangle in 3D space to form an irregular pyramid with an apex at the shoulder. Is there any way to measure volume of this irregular pyramid? Yours sincerely, Amir Date: 8/2/96 at 13:8:40 From: Doctor Tom Subject: Re: Volume of Irregular Pyramid Formed by Body I'm not sure I completely understand the description of your problem; let me re-state it to make certain I do. Your hands, shoulders, and elbows and stuff represent points in space, and you'd like the volume of the enclosed "pyramid." It sounds like the pyramid you have in mind is technically called a "tetrahedron" - a solid having a triangle as each of its faces. The Egyptian pyramids have one side that's a square and four sides that are triangles. I will answer for a tetrahedron, since more complicated solids (like Egyptian pyramids) can be broken up into tetrahedrons, and you can find the volumes of those, and add them up. The volume of any pyramid is the area of the base times the height times 1/3. For an irregular pyramid in space, these may be difficult to measure. I'll suppose that you have the 3-D coordinates of the points in space: (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), and (x4,y4,z4). If you know what a determinant is, the volume is simply this: 1 | x1 y1 z1 1 | - | x2 y2 z2 1 | 6 | x3 y3 z3 1 | | x4 y4 z4 1 | One sixth of the determinant. Depending on the order you put in the points, the number above may be positive or negative for technical reasons. If you get -3, then the volume is just +3. Expanding a determinant is not too hard -- it's just a series of additions and multiplications, but writing a complete description of the process here would be a waste of my time, since the process is described in detail in any book on linear algebra or determinants. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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