Consider any closed, three-dimensional body having volume V
and surface area A. The following inequality
always holds!
36 Pi * V2 <= A3.
It is called an isoperimetric inequality.
Presentation Suggestions:
All students "know" that the area enclosed by a plane curve
of a given perimeter is maximized when the curve is a
circle. Other closed curves of the same perimeter enclose
less area. The result quoted above can be presented in the
following way: given a closed surface of a given area,
the volume enclosed by the surface is constrained by the
isoperimetric inequality.
Students find it interesting to contrast this with the
"opposite" situation: solid bodies of a given volume V
have a minimum surface area, but not a maximum surface area!
The Math Behind the Fact:
The proof of the inequality in three dimensions is beyond
an elementary course, but it is discussed in Chapter 7 of
the Courant and Robbins reference. They give
a proof of the planar result
that does not involve the variational calculus.
The Honsberger reference gives a nice short proof of the
isoperimetric inequality in two dimensions.
How to Cite this Page:
Su, Francis E., et al. "Isoperimetric Inequality."
Mudd Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
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