Suggester shapes

Today I saw the below shape in a design on a T-shirt. It happened that at the time I was in the middle of a conversation with somebody who shares my interest in teaching math. So I said to him “that would make an interesting math problem”.

It turns out that the moment I said those words, he thought of the same math problem I was thinking of. Apparently, just saying the word “math problem” and showing this picture was sufficient to define a particular associated math problem. In this case, the problem we both thought of was:

Find a shape whose area is 3/4 the area of a square and whose perimeter is the same as the perimeter of the square.

Notice that this is more or less the same as saying “find a shape whose area is three and whose perimeter is eight”. When I rephrased it that wasy in my head, I thought of another solution:

In the above picture, I put in the dashed lines just to make it clear what’s going on — the shape is a box three times wider than it is high. Interestingly, if we had started with this shape, I doubt either of us would have come up with that particular math problem. Apparently the first shape suggests the math problem, but the second shape does not — probably because it would suggest too many math problems, and therefore no one problem in particular.

Maybe shapes can be classified by the math problem (if any) that they suggest. Many shapes work as solutions to math problems, but I suspect that far fewer work as suggesters of one particular problem.

I leave you with the following riddle: What shape, if any, effectively suggests the following math problem:

Find a shape whose area is 1/2 the area of a square and whose perimeter is the same as the perimeter of the square.