Constructing A Sierpinski Triangle
Chapter 7 Enrichment
In this activity, you will construct something called a Sierpinski Triangle and investigate its parts.
Use the equilateral triangle with side lengths of 8" in the back of this packet as your original triangle
(Stage 0). Find and mark the midpoint on each side of the triangle. Connect the midpoints with line
segments. The resulting triangle is called Stage 1.
Midpoints of Stage 0
Connected Midpoints Stage 1
1. How many small triangles do you see in stage 1? ________
2. How long will the sides of each of these triangles be?
________ Explain how you know this.
3. What kind of a triangle is each of the small triangles? _______________________
Some of these small triangles point up like the original triangle and some point down.
4. How many small triangles point up and how many point down?
______ up ______ down
5. In each of the small upward pointing triangles, how far from the ends of the sides will the
midpoints of these triangles be? ______
Find the midpoints of the sides of all the small upward pointing triangles and connect them with line
segments in each triangle. Do not do this for the small triangle pointing down. This is stage 2.
Midpoints of Stage 1
Stage 2
6. How many small upward pointing triangles can you count in stage 2? ________
7. How long is each side of these triangles? ________
Find the midpoints of all the new small upward pointing triangles and connect them with line
segments. This is stage 3.
Stage 3
8. The midpoints you just found are how far from the ends of the sides of the
triangles? _______________
9. How many new upward pointing triangles do you now have in stage 3? ________
Have you figured out the pattern? Draw stage 4.
Stage 4
10. How long will each side of the small triangles be in stage 4? ________
11. How many upward pointing triangles will you get in stage 4? ________
Shade in all the upward pointing triangles in stage 4. These are the first 4 stages in constructing what is
called the Sierpinski Triangle, named after the Polish mathematician Waclaw Sierpinski. In this
triangle, this process is supposed to go on forever, but it soon gets too small for us to see it. Can you
see that it is made up of smaller and smaller copies of itself? Objects like this are called fractals.
Shaded Stage 3
Shaded Stage 4
Name: ________________________________
Sierpinski Triangle
Sierpinski Triangle
1) Fill in the table:
Stage
1
2
3
4
n
Total Number of Upward Pointing Triangles
Length of Each Side of Upward Pointing Triangles
Area of Each Upward Pointing Triangle
Total Area of Upward Pointing Triangles
2) What patterns do you see in the numbers for the total number of upward pointing triangles?
3) Can you build a formula for the number of upward pointing triangles at the nth stage?
4) What patterns do you see in the numbers for the length of each side of an upward pointing triangle?
5) Can you build a formula for the length of each side of an upward pointing triangle at the nth stage?
6) How can you calculate the height of each upward pointing triangle if you know the side length?
7) Calculate the area of an upward pointing triangle for each of the 4 stages. Show your work:
Area - Stage 1:
Area - Stage 2:
Area - Stage 3:
Area - Stage 4: