Intermath | Workshop Support


Same Size Perimeter 

Problem Statement
Consider the shape shown below. Can you keep the perimeter the same and make the area smaller? Larger? If so, draw the new figure.

Problem setup

The challenge presents a definite shape with some alternative points. We are asked to keep the same perimeter but attempt to rearrange the construction of the line segments so that the area is reduced.


Plans to Solve/Investigate the Problem

I will begin by constructing the given figure in GeoSketchPad. I will include all of the points, so that I may be able to rearrange the line segments while keeping the integrity of the measurement of those segments. I will construct the figure by creating a line segment, and then using the Transform function to rotate the segment to the appropriate position whereby the figure will be drawn accurately and the measurement of the segment will always be the same. I will then copy the figure, and rotate selected line segments to appropriate points, whereby the figure will be changed but the perimeter will remain the same. I will then measure the areas and compare to see if the perimeter can remain the same but the area be changed.


Investigation/Exploration of the Problem

I constructed the figures in GSP as described above. The perimeters, as you can see, did indeed remain the same, but the areas changed depending on the concavity of the angle. The more angles that can be concave in, the smaller the area. See diagram below:




Extensions of the Problem


A possible extension of this problem would be to continue drawing figures with n sides; calculate the number of different figures that can be calculated, and also calculate the corresponding areas to see if a pattern develops for the relationship of the areas in reference to the length and number of sides.

Author & Contact
Jim Taylor