# Isoperimetric Inequality

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.

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 (“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.

Su, Francis E., et al. “Isoperimetric Inequality.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Courant and Robbins, What is Mathematics?
Ross Honsberger, Ingenuity in Mathematics.

Fun Fact suggested by:
Michael Moody

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