What's the largest volume that can be enclosed by a bubble of surface area A?
If V is the volume of a closed, three-dimensional region, and A is its surface area, then the following inequality always holds!
36 Pi * V2 <= A3.
This isoperimetric inequality constrains how large the volume can be.
You'll note that this inequality is maximized
when the bounding surface is a sphere!
You might also note that if V is fixed, then this inequality constrains
how small the surface area A can be.
A bubble actually tries to minimize its surface area,
which is why they tend to be spherical.
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 ("iso"-"perimeter")
enclose less area. The result quoted above is a 3-dimensional version.
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."
Math Fun Facts.