Similar triangles · Ages 14–15
The Sierpinski triangle
Each step: count × 3, area × 34, side ÷ 2.
How to do it
Method
Each step: count × 3, area × 3/4, side ÷ 2
Step 3: 33 = 27 triangles, area (34)3 = 2764
Another way
Step by step
1 → 3 → 9 → 27; 14 × 14 × 14 = 164, 27 × 164 = 2764
In full
The Sierpinski triangle is a fractal: join the midpoints of an equilateral triangle's sides to cut it into 4 equal smaller triangles and take out the middle one; then do the same to every triangle left, again and again. At each step every triangle becomes 3, each side halves, and each area becomes 14 of what it was. One way: find the pattern. Triangles left: 1, 3, 9, 27, …, so 3n after n steps. Area left: times 34 each step, so (34)n of the first after n steps. Another: step by step. After 2 steps: 3 × 3 = 9 triangles, each 14 × 14 = 116 of the first, 916 in all — the same as (34)2. Triangles taken out: 1 at the first step, 3 at the second, 9 at the third, 1 + 3 + 9 = 13 in all. The perimeter (all the triangles left, added up) is times 32 each step: three times as many triangles, each side half as long.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The picture shows one step of the Sierpinski triangle. Drawing 3 more steps from it, there are small shaded triangles.
Show the answer and working
Answer: 243
The picture has 9 shaded triangles; each step turns every one into 3: 9 × 33 = 243.
Example 2
Drawing the Sierpinski triangle: after step 4, small shaded triangles are left.
Show the answer and working
Answer: 81
Times 3 each step: 34 = 81.
Example 3
After step 2 of the Sierpinski triangle, the shaded area is less than half the first triangle's.
True or false?
Show the answer and working
Answer: False
What's left is (34)2: 34 and 916 are both over a half; 2764 is the first under it.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Drawing the Sierpinski triangle: after step 2, small shaded triangles are left.
Answer:
2.After step 2 of the Sierpinski triangle, the shaded area is of the first triangle's.
Fill in each blank
Answer:
3.Drawing the Sierpinski triangle from step 1 to step 5, triangles are taken out in all.
Answer:
4.An equilateral triangle has sides of 24 cm. Made into a Sierpinski triangle, after step 3 the perimeters of all the small shaded triangles add up to
Answer:
5.An equilateral triangle has an area of 4 cm². Made into a Sierpinski triangle, after step 1 the shaded area =
Answer:
All the answers
- 1. 9
- 2. 9, 16
- 3. 121
- 4. 243 cm
- 5. 3 cm²
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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