PITTSBURGH (KDKA) — I had a bit of a problem. I needed to fit my entire body through a hole in a standard piece of printer paper.
While this isn’t a real problem, it is a fun way to show how the perimeter and area of something is related.
Obviously, if I cut a hole in this paper, I still wouldn’t fit. This hole is not big enough, because the original shape of the paper remained the same. Even though it looks like this is not going to work out, it is possible with the right solution!
So, how can I pass through this piece of paper? Math, of course!
We are going to round down the paper size here, to make the math easier. An 8×11 inch piece of paper has a area of 88 square inches. We can’t change the area of the paper.
The perimeter of this paper is 38”. I can’t fit through that without ripping the paper, but we can add to the perimeter?
You may be thinking, “Wouldn’t that increase the area?” The answer is no, if we add more rectangles. To do that, we just need some scissors.
Fold the paper in half long ways and make a cut about an inch in on each end until you get to an inch from the other side. Do not cut all the way through! Now, about an inch away, make another cut from the other end of the paper, but stop about an inch from the other side. Keep this up until you add as many rectangles as you can on the paper. You are basically cutting a maze into the paper.
Each rectangle you make has its own perimeter. I estimate that each rectangle has 2 long sides about 2. inches long. We will multiply each of these by two, since the paper is folded in half. You can see how this perimeter can quickly add up to more than our original perimeter of 38 inches. The thinner you make the cuts, the more rectangles you will make. The more rectangles that you make, the greater the perimeter of all the rectangles will add up to!
The final step to make this work is to start one rectangle in and cut along the fold. Stop before you cut through the last rectangle.
Then you open it up and reveal a giant hole, that even an ogre like me can get through!
There is still the same area of 88 square inches of paper here, we just turned one rectangle into a bunch of them to increase the perimeter to this monstrosity. That means the area of paper does not change, even though you changed the perimeter.