Date: Tue, 6 Apr 1999 11:24:46 -0600 (CST)
Subject: Geometry

Name: Jesse

Who is asking: Other
Level: Secondary


How do I find the surface area of an irregulary shaped object such as someone's knee from the thigh to shin?


Hi Jesse,

Area, even in the plane, is really defined by three properties:

    (a) if you divide an object into several piece, the area of the original is the sum of the areas of the pieces.  
    (b) if two pieces are congruent, they have the same area.  
    (c) if one object is contained in another, it has a smaller (or equal) area).  
  In fact, if you have any ways of 'measuring' plane objects with non-negative real numbers, which satisfies these properties, then you ARE measuring area (up to some constant - the measure of a unit square). You may be familiar with these properties, because they are at the heart of AlgeTiles and other ways of visualizing algebra with geometric objects.
   Now to find the area of something irregular in 3-space, like you describe, you need to chop it up into small pieces (using principles (a), (b) and (c)) and then somehow compare the areas of the pieces with plane areas. In general, to do this without error requires cutting the object into an infinite number of pieces, and the 'sum of the areas' is an Integral (from calculus).
   You could approximate the area (perhaps give upper and lower bounds) by dividing it into a number of pieces and estimating the area of the pieces. For example, you could put down a 'grid' of points, create triangles, and then find the areas of the triangles (using straight lines in 3-space between the points - for a lower bound).
   I can't think of a simple approximation which will give you a solid upper-bound for the area of an irregular area like you have here.

Walter Whiteley
York University

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