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Convex Polygons and Other Questions

Date: Mon, 22 Jan 1996 10:35:22 -0500
From: Sweet Home Middle School
Subject: Convex polygons & other questions

Dear Dr. Math:

Our 7th grade advanced math students submitted these questions to
you last week while you were "out".  Instead of having them come up 
and retype them, I am typing them all in one message and sending them 
out to you today for them.  I hope this is ok.

        Here are their questions:

Given a fixed are, how can I maximze the perimeter of a convex 
      - Mrs. Reimer, Math teacher

How do you find the area of an oval? - Jackie, Robin & Chris

If a convex polygon had 1400 square yards, what would be the maximum 
      - Zack, Ryan & Jeff

What convex polygon has the greatest perimeter, but has an area of
1400 yds? 
      - Kristy, Mary, Jill, & Alyssa

What is the largest possible perimeter for an area of 1400 square 
      - Nicole, Katie & Kristen

        TIA for your help.

Rebecca Silverman
Sweet Home Middle School
Amherst, N.Y. 14226   716-837-3500

From: Doctor Jerry
Subject: Re: Convex polygons & other questions

These are interesting questions.

For the first, the answer is that you can't maximize the perimeter, 
given a fixed area. To illustrate this, I'll take the area to be 1 
and the convex polygon to be an isoceles triangle with base b and 
height h.

The area of this triangle is (1/2)*b*h, which we'll set equal to 1.  

The perimeter of this isoceles triangle (just use the Pythagorean 
Theorem) is 

P = 2*sqrt(h^2+(b/2)^2)+b.

Since h=2/b (from above), the perimeter is

P = 2*sqrt(4/b^2+b^2/4)+b.

If b is taken to be a very small number (which forces the triangle to 
be a very narrow but very high triangle), the perimeter becomes large.  
If, for example, b = 0.01, then P = 400.01....  
If you take b even smaller, the perimeter is even larger.  
So, the perimeter can not be maximized. 

For the area of an oval, an equation describing the oval would have to 
be given.  For ovals which are ellipses, there is a known formula.  
A circle is a special case of an ellipe.  Its area of pi*r^2, where 
r is its radius.  Ellipses have two radii, as it were, a shortest 
radius and a longest radius.  If these are b and a, then the area of 
an ellipse is pi*a*b.  If a = b, then we get the area of a circle.

Questions 3, 4, and 5 appear to be the same as the first question.  
There is no maximum.  My answer was in terms of triangles.  A similar 
thing happens for other convex polygons.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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