 My Math Forum Proving The Maximum Area of a Shape Regardless of How Many Sides It Has

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 December 30th, 2017, 05:09 PM #1 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 675 Thanks: 88 Proving The Maximum Area of a Shape Regardless of How Many Sides It Has The greatest area of a quadrilateral given a fixed perimeter is a square. This seems to be true for an equilateral triangle. Is there a proof that given a fixed perimeter and fixed amount of sides, the area will be maximized when all the sides are equal regardless of how many sides there are? December 30th, 2017, 06:00 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 684 Thanks: 459 Math Focus: Dynamical systems, analytic function theory, numerics I'm a bit short on time atm but here is the idea. Suppose a $n$-polygon is specified as a list of $n+1$ vertices, $\{(x_0,y_0),\dotsc,(x_n,y_n)\}$ and let $\gamma_k$ denote the line segment between the $(k-1)^{\rm st}$ and $k^{\rm th}$ vertex. Then the area is given explicitly by the line integral $\sum_{k=1}^n \int_{\gamma_k} x \ dy$ Now, regard this as a function of $2(n+1)$-many variables and maximize it using standard multi-variable optimization techniques. Thanks from Maschke December 31st, 2017, 01:46 PM #3 Global Moderator   Joined: May 2007 Posts: 6,855 Thanks: 744 For any figure more than 3 sides, it is important to include a requirement that all angles be equal. Example for 4 sides: a rhombus has all sides equal, but it can be squeezed to an area as close to 0 as one wants. Thanks from Country Boy January 4th, 2018, 05:19 PM   #4
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 Originally Posted by Knowledgesearcher I got a lot of answers there.
Oh, I'm sure you did.. Tags area, maximum, proving, shape, sides Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post bparker Applied Math 1 April 19th, 2015 11:27 AM matisolla Calculus 2 February 4th, 2015 01:49 PM mathismydoom Algebra 6 September 29th, 2012 06:34 AM NASAorbust Algebra 15 March 21st, 2011 09:50 AM telltree Algebra 0 January 21st, 2010 01:51 PM

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