Surface area of humans
Name: betty a laliberte
Date: Around 1995
We want to learn how much skin our classmate Steve has. We have been
thinking of dividing the body into geometric shapes. Our problem is that
the body seems to be oddly shaped. We have tried calling doctors and
nurses, but they are not too interested in our problem. Would you help?
The point of the question is not the numerical answer. It is, how do you go
about finding the answer? I think that the question is a very good one, it
is designed to get you to THINK (not just repeat someone else's answer), and
I encourage you to explore possible answers with each other. In the
meantime, please congratulate your teacher for me.
That IS an interesting problem! I do not know exactly how to approach it.
It may keep me up all night. One thought I have is that you might be able
to exploit similarity somehow. If you can find a toy model (e.g. a "Ken"
doll or a GI Joe) that is approximately similar to Steve, then if you can
answer the question about the doll you can use ideas of similarity to
estimate an answer for Steve.
The advantage of this would be that you can perform experiments on the doll
that Steve might object to (e.g. wrap it in silly putty). I expect it would
be easy to find Steve's volume that way. Surface area will require more
creativity. What could you do with a
photo of Steve?
Well, this is essentially a problem in "surveying." How do surveyors figure
out distances and areas etc.? You need to set up some kind of grid or
collection of points on the body, in between which the surface is close to
flat. The other option, of course, is the physicists approach - assume
Steve's surface is fractal, estimate fractal dimension, calculate a lower
bound from his volume (and the surface area of a sphere of the same volume)
and go from there. (The surface area of a fractal body like that should be
infinite so you will have to cut it off at some practical dimension - say
the size of a finger, or maybe the width of his hairs - assuming you were
planning on counting his hairs, right?)
1) Cover him with triangles or some other regular figures. If the figures
are small enough (and the guy is smooth) you will get a good approximation
to his surface areal by adding the areas of the figures that cover him.
2) If you like to be more physical, you can cover him with paint; make sure
the layer is equally thick everywhere. Divide the volume of paint used by
the thickness of the paint layer - this ought to be the surface area or a
My son had a different idea that I thought was workable. Approximate
Steve's body with a bunch of solid shapes for which you know surface area
formulas (cylinders, spheres, and so forth). For example, his head might be
spherical of ellipsoidal, his neck a cylinder, etc.
Problem 27 on page 832 of CALCULUS (3e) by D. G. Zill quotes the approxima-
tion in terms of weight, w, and height, h, as
S = 0.1091*(w**0.425)*(h**0.725). Worth a try!?
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Update: June 2012