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.
How to Cite this Page:
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