Math Forum - Project of the Month

January 1998 Project of the Month

Our guest problem-poser this month, Joshua Zucker, is a precalculus and calculus tutor at the Education Program for Gifted Youth, Stanford. He is also currently developing a computer-based geometry course for that program. See http://www-epgy.stanford.edu/epgy for more about the program, http://www-epgy.stanford.edu/epgy/nocd/tutors/josh.html for more about Josh.
You need to do as much of the first two parts as you can. The second two parts are a little extra, in case you get really interested in the problem!

I was looking at a common right triangle the other day, and noticed that when I computed its perimeter, I got the same number as when I computed its area (though of course the units were different, being centimeters for perimeter and square centimeters for area).

Part 1

(a) What right triangle might it have been?

(b) Given that the triangle had integer sides, what are *all* possible triangles?

(c) If you're not given that the sides were all integers, what relation(s) can you find among the sides? Are there any limiting special cases?

Part 2:

After I finished thinking about right triangles, I got to wondering about whether this property (of having perimeter in centimeters equalling area in square centimeters) was possible for other types of triangles.

(a) Are there isosceles triangles with integer sides that have this property? If so, find them. If not, explain why not.

(b) What if just the two legs are integers? Or if just the base is an integer?

(c) What relations are there between the legs and the base? Are there any special limiting cases?

Part 3:

What about general triangles? I thought about Heron's formula for the area, and the simple a + b + c for perimeter, and tried to work from there, but I didn't get very far. All I know is a cubic equation relating a, b, and c.

(a) Are there any other triangles with integer sides that have area numerically equal to perimeter (that are not covered in the right and isosceles sections above)?

(b) Can you find any other relations between the sides that are simpler to solve than the cubic equation that comes from setting a + b + c equal to Heron's formula for the area?

(c) Write your own questions about these triangles (with area numerically equal to perimeter) and, if possible, answer them.

Part 4

How about quadrilaterals? The 4x4 square certainly works, and no other squares.

(a) How about rectangles?

(b) Parallelograms?

(c) Why is this so much easier than the triangle problem?

(d) What constraints are there? That is, describe a class of parallelogram side lengths for which the area CANNOT equal the perimeter.