Equal Area and Perimeter

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

Consider convex polygons drawn in the plane. A fixed polygon will have an area, a perimeter, and a number of sides. The purpose of this microworld is to explore to what extent different polygons with the same area and perimeter can differ in the number of sides they have. Quite a bit is known about problems of this kind but I give no references so that the excitement of seeing what can be discovered on one's own is present. However, the questions relating to equiangular and equilateral polygons seem to be "new."

1. Given a pair of integers m and n (each at least 3), for what pairs (m,n) can one have a pair of not-congruent convex polygons P and Q where P has m sides and Q has n sides so that P and Q have the same perimeter and area?

2. Given a pair of integers m and n (each at least 3), for what pairs (m,n) can one have a pair of not-congruent convex polygons P and Q where both polygons are equilateral and P has m sides and Q has n sides so that P and Q have the same perimeter and area?

Note: It may be of interest here to consider this question. If P is an equilateral convex m-gon, what is the largest area P can have and what is the smallest area that P can have? (If a max or min can't be obtained, you can consider the least upper bound and greatest lower bound for the numbers involved.)

3. Given a pair of integers m and n (each at least 3), for what pairs (m,n) can one have a pair of not-congruent convex polygons P and Q where both polygons are equiangular and P has m sides and Q has n sides so that P and Q have the same perimeter and area?

Note: It may be of interest here to consider this question. If P is an equiangular convex m-gon, what is the largest area P can have and what is the smallest area that P can have? (If a max or min can't be obtained, you can consider the least upper bound and greatest lower bound for the numbers involved.)

4. Given a pair of integers m and n (each at least 3), for what pairs (m,n) can one have a pair of not-congruent convex polygons P and Q where P is equiangular and Q is equilateral, and P has m sides and Q has n sides so that P and Q have the same perimeter and area?

5. Does anything interesting happen if we consider the above questions for simple non-convex polygons? Can equiangularity occur? If not what might be reasonably substituted to result in interesting questions?

6. When two polygons have the same area it is known that that one can cut up one of the polygons into a finite number of polygonal pieces and reassemble the pieces to form the other polygon. It might be of interest to look at minimal dissection solutions for pairs (m, n) as discussed above.