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 polygon? - 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 perimeter? - 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 yards? - Nicole, Katie & Kristen TIA for your help. Rebecca Silverman Librarian 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. So, 1=(1/2)*b*h. 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/
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