What’s the largest volume that can be enclosed by a bubble of surface area A?<\/p>\n\n\n\n
If V is the volume of a closed, three-dimensional region, and A is its surface area, then the following inequality always holds!<\/p>\n\n\n\n
36 Pi * V2<\/sup>\u00a0<= A3<\/sup>.<\/p>\n\n\n\n This\u00a0isoperimetric inequality<\/em>\u00a0constrains how large the volume can be. You’ll note that this inequality is maximized when the bounding surface is a sphere!<\/p>\n\n\n\n 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\u00a0surface area, which is why they tend to be spherical.<\/p>\n\n\n\n Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by: <\/strong> What’s the largest volume that can be enclosed by a bubble of surface area A? If V is the volume...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[8,4],"class_list":["post-411","page","type-page","status-publish","hentry","tag-geometry","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/411","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=411"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/411\/revisions"}],"predecessor-version":[{"id":1492,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/411\/revisions\/1492"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=411"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
Su, Francis E., et al. “Isoperimetric Inequality.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Courant and Robbins, What is Mathematics?<\/a><\/em>
Ross Honsberger, Ingenuity in Mathematics<\/a><\/em>.<\/p>\n\n\n\n
Michael Moody <\/p>\n","protected":false},"excerpt":{"rendered":"