There are other situations that involve both area and perimeter. Consider this one: Joel, a student in a sixth-grade class, completed an exercise similar to one in Session 6. After tracing his hand on grid paper, he was asked to approximate its area. Instead of counting squares, he took a piece of string and traced the perimeter of his hand. He then took the length of string that represented the perimeter of his hand, reshaped it into a rectangle, and found the area of the rectangle. Joel concluded that the area of his hand and the area of the rectangle were the same.
Think about Joel's strategy and conclusion. Will his strategy work in all situations? Do you agree or disagree with his conclusions?
Think about what you discovered in Problems A1 and A2 and how that information might be used to analyze Joel's strategy. Close Tip
Looking at this situation from another direction, do figures with the same area always have the same perimeter? Why or why not? And if not, which perimeters are possible, and which are impossible?
Use this Interactive Activity to investigate this question. You can arrange 12 square "tiles" to make plane figures like the ones below. Each square must share at least one side with another square.
This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. For a non-interactive version of this activity, use 12 one-inch square plastic or ceramic tiles (or Scrabble tiles).
What is the smallest perimeter possible using 12 square tiles?
What is the largest possible perimeter?
Make figures with an area of 12 square units with perimeters of 14 through 26 units. Keep in mind that it is not possible to create all of these perimeters. Sketch the shapes and record their areas and perimeters.
Choose one perimeter between 14 and 26 that you could not make and explain why it is impossible.
Under what circumstances might you want the smallest perimeter for a set area?