We have two rugs, one is square and the other is rectangle.
Can we compare them to find out which one has the larger area, but without measuring their distances?
If they are made of the same material we can weigh them to get a estimate of the difference in area.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
No, it must be a geometric solution but without measuring - only by comparing.
I was thinking something like this:
If one shape fits perfectly inside the other one, then it is easy. Same also if one of the two dimensions are exactly the same (for example if the square is AxA and the rectangle BxC and A=B or A=C, then we compare the other sides only.
If, however, say, A>B and A<C then I guess we put one rug onto the other so that they have one common vertex (say, the lower left vertex) and then what?