Rectangle Puzzle: Area = Perimeter
Every square has a perimeter, and an area. If the perimeter is 12, the area is 9 (check that).
But one special square has its area equal to its perimeter. (Answer is in a few lines).
You could just guess at it, and get it. You could do some algebra:
And yeah, the area of a square with side 4 is 16, and so is the perimeter.
Rectangles (here’s the puzzle)
Now, rectangles come in more varieties than squares. That can make them more interesting, or more complicated.
Can a rectangle have its area equal its perimeter? Yes, 4×4 works (remember, a square is a special rectangle). But there is at least one more.
Can you find another rectangle with its area equal its perimeter?
Can you find all of them?
How do you know when you have them all?
JD2718 
I encourage you to take this a little farther, and challenge them to prove that any closed squiggle has a similar image that has perimeter and area equal. A rectangle, for example, that is 2 by 8 has a perimeter of 20 and an area of 16. If I multiply all its sides by 2, then the perimeter is 40, and the new area is 64. Our area was smaller in one case, larger in the other. Surely a multiple of the first between 1 and 2 will make them equal. This growth relation between similar objects of similar shape is a part of a major geometric relationship between shapes, as you well know. Expose students to it at every opportunity.
This is interesting, and the timing is fortuitous.
I did not present this in class recently – I’m aiming to intersect with James Taunton’s June work – I got the gist on twitter, and he showed me where he’s going.
But – that’s a great angle you’ve presented. I need to find a way to share it with students.
I love the nerding out. I’m an engineer and just came across this. Clearly you are committed to your craft!