Regular and Non-regular Polygon AreasDate: 03/10/99 at 17:10:37 From: Robert Davies Subject: Regular Polygons I am trying to work out the proof that regular polygons give the maximum area but as of yet have not succeeded. Please help! Date: 03/11/99 at 11:55:35 From: Doctor Peterson Subject: Re: Regular Polygons The first approach that comes to mind is to take any non-regular polygon and show that you can find a larger polygon with the same perimeter. Then the largest polygon with that perimeter must be regular. Suppose that there are three consecutive vertices A, B, and C in a polygon such that AB and BC have different lengths. See if you can find a point B' where AB' and B'C are the same length, but their sum is the same as AB + BC. Then show that the area of triangle AB'C will be larger than that of ABC. If you replace B with B' in the polygon, its perimeter stays the same but the area is larger. B' B + + / \ / \ / \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ A // \\ C + - - - - - - - - - - - - - - - - - + | \ | \ ... ... This will show that the largest polygon has to have all sides the same, since a polygon whose sides are not the same is never the largest. You will also have to show that the angles in the largest polygon with a given perimeter are all the same. Try a method similar to what we just did for the sides. (I have not taken the time to work that part out.) - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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