> How many right triangles, with integreal sides, have their areas = to > their perimeters? How many rectangles with integral sides have this > same property? Is there a gemerally accepted name for any geometric > figure which has this property. (The circle with radius 2 is another)
Answer: Not very many. Actually, only two such triangles exist: the 5-12-13 triangle and the 6-8-10 triangle. The first one has an area and a perimeter of 30, and the second one has an area and a perimeter of 24.
For rectangles, there are also only two of them: the rectangle with sides of 3 and 6 (area & perimeter of 18) and the 'rectangle' (square) with sides of 4 (area & perimeter of 16).
How do I know? Actually, I don't know quite for sure. Just for the heck of it, I wrote a couple of little Pascal programs that checked all the cases from sides of 1 up to 100 or 200, respectively. (Amazing how fast they run on a Pentium, instead of an 8088!--took less than a second).
I imagine one could write a proof of my conjecture, however.