If you have 72 ft. of fencing and you want to use it to make a rectangular pen for your Highland terrier, you must consider both the perimeter of the pen and its area. What relationships exist between these two measures? Do shapes with the same perimeter have the same area? Let's investigate this situation.
Imagine that you want to use all 72 ft. of fencing to make the rectangular pen, that the dimensions of the pen in feet will be whole-number values, and that you want the maximum area for your puppy.
What are the dimensions of the possible rectangular pens?
What are the areas of these pens?
All of the pens have a perimeter of 72 ft., yet the areas of the pens differ. What do you notice about the shapes of the pens with small areas as opposed to those with large areas? What are the characteristics of a shape with the greatest area?
In the above problems, you've seen that when you form the fencing into a long, skinny rectangle, the area is small. But the area increases as the rectangle becomes more square-like, and the greatest area occurs when the fencing is in the shape of a square or square-like rectangle. This leads us to consider shapes other than rectangles. For example, if the perimeter remained the same, would an equilateral triangle or a regular pentagon or a regular hexagon have the same area or more or less area than the square?
Imagine the pen were in the shape of an equilateral triangle. What is the area of this triangular pen?
Draw a picture of a triangle. How long is each side? Draw the height of the triangle, and use the Pythagorean theorem to find the height. Close Tip
Imagine the pen were in the shape of a regular hexagon. What is the area of this hexagonal pen?
Draw a picture of a hexagon. What is the length of each side? You can divide the hexagon into six equilateral triangles and calculate the area of a single triangle using the method from Problem A3. Close Tip
Would other shapes give the puppy even more square footage? Imagine building a circular pen. Find the area when the circumference is exactly 72 ft.
In the activities above, you've seen that when the perimeter is fixed, shapes that have many sides have a greater area. In fact, the shape with the greatest area when the perimeter remains constant is a circle.
Imagine you have a barn that is 70 ft. long on your property. You plan to use a part of the existing barn wall as one side of the fence. What are some options for the shape of the pen? What shape of the pen will give the greatest area under these conditions?
Consider several different shapes for the pen -- square, rectangle, semicircle -- and collect data for each of them. Close Tip
Video Segment In this video segment, David Cellucci and David Russell examine how to maximize the area of the puppy pen built against a barn. They try several different shapes, including rectangles and semicircles. Watch this segment after you've completed Problem A6.
Were your findings similar?
If you are using a VCR, you can find this segment on the session video approximately 4 minutes and 56 seconds after the Annenberg Media logo.