
Geometry Geometry Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 30th, 2017, 04:09 PM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 602 Thanks: 82  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, 05:00 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 383 Thanks: 207 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 $(k1)^{\rm st}$ 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 multivariable optimization techniques. 
December 31st, 2017, 12:46 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,528 Thanks: 589 
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. 
January 4th, 2018, 04:19 PM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,597 Thanks: 546 Math Focus: Yet to find out.  

Tags 
area, maximum, proving, shape, sides 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How do I determine how many sides a shape has?  bparker  Applied Math  1  April 19th, 2015 10:27 AM 
Area of shape formed by circles  matisolla  Calculus  2  February 4th, 2015 12:49 PM 
I have a question about the area of this shape?  mathismydoom  Algebra  6  September 29th, 2012 05:34 AM 
Area for this shape  NASAorbust  Algebra  15  March 21st, 2011 08:50 AM 
Det. length of sides of polygon from Area and ratio of sides  telltree  Algebra  0  January 21st, 2010 12:51 PM 