From: Ronald Bruck Subject: Re: Area of regular polygon vs. irregular Date: Mon, 13 Mar 2000 19:01:15 -0800 Newsgroups: sci.math Summary: [missing] In article <8ak1kh$ahf@euclid.ics.uci.edu>, eppstein@euclid.ics.uci.edu (David Eppstein) wrote: > desther4@yahoo.com writes: > > Why must the area of a regular polygon (of n sides) alway sbe greater > > than the area of an irregular polygon (of n sides)? > > It mustn't. It is easy to find irregular polygons with area (say) an > acre, and regular polygons with the area of a pinhead. Perhaps you can > phrase your problem more carefully. An even more amazing fact: the polygon of a given DIAMETER which has maximum area need not be regular! There's a Java script at which illustrates this for the hexagon. It doesn't need illustrating, of course; the hexagon was solved by Ron Graham. But I don't believe the optimal octagon has been proved correct (Graham conjectures the general solution). Note that "of a given diameter" and "which can be inscribed in a circle of a given diameter" are not the same thing. --Ron Bruck -- Due to University fiscal constraints, all .sigs must be only one line.