Cynthia Lanius

The Koch Snowflake

What about the AREA?

Table of Contents    Introduction   Why study fractals?     What's so hot about     fractals, anyway?   Making fractals     Sierpinski Triangle          Using Java          Math questions          Sierpinski Meets Pascal     Jurassic Park Fractal          Using JAVA          It grows complex          Real first iteration          Encoding the fractal          World's Largest     Koch Snowflake          Using Java          Infinite perimeter          Finite area          Anti-Snowflake             Using Java   Fractal Properties     Self-similarity     Fractional dimension     Formation by iteration   For Teachers     Teachers' Notes     Teacher-to-Teacher   Comments     My fractals mail     Send fractals mail   Fractals on the Web     The Math Forum   Other Math Lessons     by Cynthia Lanius

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? If you'd like to draw and experiment with the figures, you may print and use this triangle grid paper to help you with the drawing. Remember the process: Divide a side of the triangle into three equal parts and remove the middle section. Replace the missing section with two pieces the same length as the section you removed. 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.

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.

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 go above it.

Copyright 1996-2007 Cynthia Lanius