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Topic: Areas = to Perimeters
Replies: 8   Last Post: Dec 8, 1998 3:03 PM

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 Guy F. Brandenburg Posts: 333 Registered: 12/3/04
Re: Areas = to Perimeters
Posted: Sep 26, 1998 10:47 AM

Richard Beeler wrote:

> 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.

Thanks for the question.

Date Subject Author
9/26/98 Richard Beeler
9/26/98 Guy F. Brandenburg
9/26/98 John Conway
10/8/98 Uday
9/26/98 John Conway
9/26/98 John Conway
9/26/98 Alan Lipp
9/26/98 Joshua Zucker
12/8/98 .