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).