Exploring the Relationship Between Perimeter & Area
Shelly Berman shelly@mathforum.org
Summer Session 2001
The following exchange between three students was overheard in some ideal math lab environment:
Adam: Hey, I have an idea about a relationship that the teacher didn't tell us but that seems right to me. If you look at this illustration I made, you'll see a square with a side of 4 inches and a rectangle with sides of 4 inches and 6 inches. From these, I think the bigger the perimeter of a closed figure is, the bigger its area is.
Bobby: Hmmm. Let me play with that idea using these square tiles. I'll start with a big square like you did with 4 tiles on each side. Then I'll see what other rectangles I can make.
Charlene: Gee, I'd like to get in on this too, but I'm going to start with a string. First I'll make a loop and tie it so that it just fits around Bobby's square. I'll take the loop off and stretch it around my thumbs and fingers to make different rectangles.
QUESTIONS
 Do you think the students will have the same conclusions about the relationship between perimeter and area?
 How would you characterize the difference between the explorations suggested by Bobby and Charlene?
 If these were students in your class, how would you suggest they organize and represent any results of their explorations?
 Do you think what Bobby and Charlene did tested what Adam said?
 If you were working with Adam, what would you say to him about his conjecture?
 Both Bobby's and Charlene's explorations seem to show the same thing about the relationship between the perimeter and area of rectangles. What is it?
 Use the idea developed in the previous question to help think through a solution to these problems:
A. Your math class is asked to design a sandbox for the playground. You will be using lengths of 2x6 lumber to make the outside walls of the sandbox. We are trying to get the most from what we have, so we have to use the four pieces of lumber the custodian had. They have lengths of 3 feet, 8 feet, 10 feet and 11 feet. How should we cut them up so that we can get a rectangular sandbox with the biggest area?
B. Your class did such a good job with the first sandbox that the Principal asked them to make a second one for the Kindergarten. This time, she wants the sandbox to be triangular, so the little ones can experience different basic shapes. How should the class cut up two 24 foot lengths of lumber to outline the triangular sandbox so that it will give the most play space?
C. Uh oh. Before you could get to the custodian with your design, he had already cut one of the boards in two pieces, and cemented an 18 foot length in place as one of the sides of the triangle. How should the remaining boards be used to get the most area possible?
 There are other mathematical issues imbedded in what Adam said  for example, he commented on "closed figures". If we didn't restrict ourselves to rectangles, we could explore other interesting possibilities. See what you can find out about the "Top Hat" curve or the Koch Snowflake, and describe what this means about the relationship between perimeter and area.
 We could extend Adam's conjecture a bit by adding a new dimension, to explore the relationship between area and volume. The classic problem is to make a "tray" from a standard 8.5 inch x 11 inch piece of paper by cutting congruent squares from each corner, and folding the remaining sides up. What size square should be removed from each corner to maximize the volume of the tray?
Links to Related Lesson Ideas
 Buying Tile
 Cooperative group activity designed to reinforce the concept of area. Two examples of student work are available.
 Fractals
 Includes definitions and descriptions of fractals, links to pages on the Sierpinski triangle, the Koch edge, the Peano curve, the Lorenz attractor, and the Dragon curve; and more links to fractal sites on the Web.
 Project Interactivate
 Geometry and Measurement Concepts: Includes basic notions of lines, rays and planes, area, perimeter and working with tessellations, fractals.
 Quadrilaterals
 A kinesthetic activity designed to develop group interaction and cooperation while working with constructing a large parallelogram, square, rectangle, rhombus and trapezoid using rope held by the participants.
 What is Area?
 Collaborative group activity comparing area and perimeter using HandsOn Math software by Ventura Educational Systems but the ideas could be adapted to use with other software or a Java applet.
