Question

Which geographical shape gives maximum area for given perimeter?

Answer 1

Date: 06/02/98 at 16:42:11 From: Brad Morris Subject: Calculus/geometry max/min problem Hi Math Doctor! I am a math teacher at Friends Central School in Wynnewood PA. In our Geometry book (USCMP) it states that the geometric figure of maximum area and given perimeter is a circle. In the margin it states that Calculus is required to prove it. I'm the Calculus teacher and our Geometry specialist asked me about it and I really couldn't come up with a proof. I need help! Thanks. Brad Morris Date: 06/02/98 at 18:54:43 From: Doctor Anthony Subject: Re: Calculus/geometry max/min problem We can start by first showing that for, say, a rectangular shape of given perimeter, the square gives the greatest area. If the perimeter is P, let length be x, then width is (1/2)(P-2x) and area is: A = (1/2)x(P-2x) = Px/2 - x^2 dA/dx = P/2 - 2x = 0 for max or min So 2x = P/2 and x = P/4. Thus the square is the optimum shape for a 4-sided figure. This can be extended to figures with more sides, and it is always true that a symmetrical shape will enclose the greatest area for a given perimeter. If we have a circle with radius r the perimeter is 2*pi*r. If this is also the perimeter of a square, then each side of the square has length 2*pi*r/4 = pi*r/2 and area of square is: (pi*r/2)(pi*r/2) = (pi/4)pi*r^2 = pi/4 * area of circle and since pi/4 < 1 the area of the square is less than the area of the circle. A similar exercise with a 6-sided figure gives: area of hexagon = sqrt(3)pi/6 * area of circle = 0.9068 * area of circle If we then go to an n-sided figure with perimeter P, the area of one of the basic triangles formed by joining the midpoint of the figure to the vertices is: (1/2)r * cos(pi/n) * P/n (here r = distance from centre point to a vertex), and with n such triangles the area is (1/2)r * P * cos(pi/n). This will tend to its greatest value as cos(pi/n) -> 1 which requires that n -> infinity. So we end up with a polygon of an infinite number of sides, which is a circle.