The interactive sketches on this web page can be used to investigate the following problem:
You want to create an enclosed play area for children and you have exactly 320 feet of fence. What shapes and sizes of play areas can you make?
There are an infinite number of solutions; however, discussion here is limited to circular areas and polygonal areas with whole unit side lengths. To get a feel for the problem, let's begin by considering triangles. In the sketch below, drag points B and C on segment AA to change the shape of the triangle. Notice that the lengths of the sides of the triangle change, but if you add the measures of the sides, the total remains 320 units. The sketch allows you to examine all triangles with perimeter 320 units and integral side lengths.
Click here for further mathematics related to the triangle sketch
Next, we consider a rectangle with perimeter equal to 320 units. In the sketch below, drag point A to change the length and width of the rectangle. Notice that the side lengths change, but the sum of the side lengths (AB+BC+CD+DA) remains constant. For what rectangle is the area maximized? What function describes the relationship between the lengths of the sides and the area of the rectangle? Is the function linear?
Click here for further mathematics related to the rectangle sketch
Drag points on the segment and points C and D on the pentagon below to change its size and shape. What is the largest area you can find with perimeter equal to 320 units? What is the shape of the pentagon? What are the side lengths?
Click here for further mathematics related to the pentagon sketch
Find the largest area possible for the hexagon and octagon below. Compare your findings for the pentagon, hexagon, and octagon. What patterns do you notice in the side lengths, areas, and shapes?Based on your investigations so far, what predictions can you make about the shapes, side lengths, and areas of decagons (10 sides) with perimeter equal to 320 units?
A circle with circumference equal to 320 units is shown below. There is only one such circle and it has the largest area of all!
Click here for further mathematics related to the circle sketch
References for further reading about area/perimeter relationships:
Blackwell, W. (1984). Geometry in architecture. Berkeley, CA: Key Curriculum Press. [See Chapter 2]
Courant, R., & Robbins, H. (1996). What is mathematics? 2nd edition revised by I. Stewart. New York: Oxford University Press. [See the isoperimetric problem in Chapter 7, maxima and minima.]