Volume of Irregular Pyramid of Human Arm

Date: 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
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