Cynthia Lanius

The Koch Snowflake

What about the AREA?

In the Koch Snowflake, an infinite perimeter encloses a finite area. The perimeter of the Koch Snowflake gets bigger and bigger with each step. But what about the area? Imagine drawing a circle around the original figure. No matter how large the perimeter gets, the area of the figure remains inside the circle. Could we actually calculate the area?

Remember the process:

  1. Divide a side of the triangle into three equal parts and remove the middle section.
  2. Replace the missing section with two pieces the same length as the section you removed.
  3. Do this to all three sides of the triangle.

Let's investigate the area below.

Notice the second iteration of the Koch Snowflake above. Notice that the original triangle (yellow) is still contained in the Koch Snowflake with three smaller triangles (red) added in the first iteration, and twelve even smaller triangles (blue) added in the second iteration. So finding the area of the Koch Snowflake is just an addition problem. You find the area of the original triangle, add the area of the three red triangles for the first iteration, and add the twelve blue triangles for the second iteration.

Let's use the triangles of our grid to measure the area. The original yellow triangle has 81 triangles inside. So we'll say its area is 81. What is the area of each red and each blue triangle? Let's organize all our data into a table.

Iteration No.Area of 1 triangleNo. of triangles
added
Amount of area
added
Total Area
--
   
81
1
9
3
27
108
2
1
12
12
120
Each "Area of one triangle" is 1/9 the previous one. Each "No. of triangles added" is 4 times the previous one. Let's predict the next steps from the rule we notice above.
Iteration No.Area of 1 triangleNo. of triangles
added
Amount of area
added
Total Area
--
   
81
1
9
3
27
108
2
1
12
12
120
3
1/9
48
5.33
125.33
4
1/81
192
2.37
127.7
5
1/729
768
1.05
128.75
6
1/6561
3072
.4682
129.21
Observe what is happening. The Snowflake is still growing in area but more and more slowly. It is converging on some number, getting closer and closer to that number, but will never reach it or go above it.

Let's find the number that the area is converging to. Calculate the next 4 steps of the table.

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Copyright 1996-2007 Cynthia Lanius

URL http://math.rice.edu/~lanius/frac/ko3pr.html