Perimeter/Area Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . name(s)_______________________________________

The town of Lakeview is designing a children's playground. The parents feel that the playground must be completely enclosed, for the safety of the children. A parent in the community is willing to donate 400 feet of fencing material. The community would like the playground to have the largest area possible. What shape should the playground be?

The question, stated in mathematical terms, is "What geometric shape has the greatest area for a given perimeter?", or, "What shape maximizes area with the minimum perimeter?"

Explore this concept, using the Geometer's SketchPad. For each of #1-3 below, use a New Sketch on SketchPad.

1) Begin with triangles. Draw any triangle, select the vertices and construct the Polygon Interior, using the Construct menu. Click on the interior, and measure the Perimeter and the Area using the Measure menu. Drag vertices around until the perimeter is as close to 400 (or a convenient multiple of 4) as you can make it. Note the area.

Draw a triangle with a different shape, and do the same thing: set the perimeter as close to 400 (or the multiple of 4 you used before) as you can make it and note the area. In each case, type comments and observations.

Continue to explore triangles. Does symmetry make a difference? Write a conclusion: with a given perimeter, what type of triangle has the greatest area? Check Print Preview (in the File menu) and make sure it all fits on one page by dragging things closer together if necessary; do NOT Scale to Fit Page. Type your name(s) and print one page with all your triangles, all the data on perimeter and area, and your comments and observations.

2) Now explore quadrilaterals. Again, draw a random quadrilateral: begin with a non-convex quadrilateral (one with angles that "go in"). Make the perimeter 400 (or as above) and note the area. Consider a "normal" quad, then a parallelogram, etc. Does symmetry make a difference? Write a conclusion: with a given perimeter, what type of quadrilateral has the greatest area? Print all your quadrilaterals, all the data on perimeter and area, and your comments and observations as before.

3) Now explore figures with more than 4 sides. Draw a 5-sided polygon (does symmetry matter?), and at least two more polygons, increasing the number of sides. Keep the perimeter 400 (or as above) and note the changes in area. Write a conclusion: with a given perimeter, what type of geometric figure has the greatest area? Print all your figures, all the data on perimeter and area, and your comments and observations as before. What final conclusion(s) can you make about this entire investigation?

What other applications in "real life" situations might these ideas apply to? Consider the kinds of decisions that package designers must make, in desiging cost-effective packages for commercial products such as sodas, canned food, or other household products.