Math CentralQuandaries & Queries


Question from Kelly, a parent:
Why do objects w/ the same exterior linear feet have different interior sq ft?
I.e. a 25x25 room has 625 sqft while a 20x30 room has only 600 sqft?
I can visualize that by cutting and moving parts of the rectangle,
you loose a 5x5 section, but I just can't get my head around why.

We have two responses for you

Hi Kelly.

Take a look at

It shows how an object with a constant exterior linear length (perimeter) can go from substantial area to zero area depending on its shape. We put this video clip together because many people think that by simply measuring the side lengths of a lot, they can know the area. But of course, the angles matter too.

Hope this helps,
Stephen La Rocque


In math, the question is "why should it be true?" But if we can settle whether something is true or false at all, then it is only true if we can show a reason that works for all cases. For it _not_ to be true a single counterexample - such as the one you gave - is sufficient.

In fact, cutting pieces and moving them around furnishes a valid way to prove the areas are different. That was actually the definition of equal/unequal area that Euclid used, and it's a very good one - though it doesn't generalize well to higher dimensions. So the answer is: they are not in general the same, and you know this because in at least one case you have proved it.

A way to get a better intuitive feel for this is to consider extreme cases. Think about a 49x1 room, or even a 50x0 room in which the walls meet!

There are some deep philosophical questions involved here, it is true - for instance, some things _are_ just true by accident. [Example: the number of regular convex polyhedra is the same as the number of ways to arrange the numbers {1,2,3} so they aren't in increasing order - five in each case.]

Good Hunting!

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