This project is the 2nd of two projects on the Koch snowflake. The reason for the projects was that my younger son wondered how the Koch snowflake could have an infinite perimeter but a finite area.
The first project (about the perimeter) is here:
Exploring the perimeter of the Koch snowflake
Our approach to studying the area was similar to the approach for studying the perimeter. Essentially we looked at the steps in the construction of the Koch Snowflake and then looked for a pattern. Here are the initial thoughts from the kids about the area:
The first step in studying the area was to look at the total area of the first few iterations of the Koch Snowflake.
I decided to avoid the complexity of geometry triangle formulas and just talked about scaling. My younger son also came up with a really nice argument for
Now that we’ve seen a first few cases, can we find the pattern?
The amount of area that we add each time has a fairly simple pattern – it is just multiplication by 4 and division by 9. The only time that doesn’t happen is in the first step.
Can we connect the numbers with the geometry?
Now that we’ve seen and understood the pattern, how can we figure out the sum? I love that the boys saw that the main sum we were looking at here was less than 2.
I didn’t want to derive the geometric sum formula, so I just gave it to them. We can talk about it another time. That formula seems to be the easiest way to find the exact value of the sum, though.
Finally we wrapped up and discussed the process we used to study the area and perimeter. I don’t really believe that my younger son now understands every detail of what we talked about, but I hope that he’s a little bit less confused about the area and perimeter of the Koch snowflake.
I think the math here is something that all kids would find interesting.