Geometry and proof · Ages 14–15

The Sierpinski triangle worksheet

Part of: Similar triangles

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Geometry and proof · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.The picture shows one step of the Sierpinski triangle. Drawing 3 more steps from it, there are small shaded triangles.
Answer:
2.Drawing the Sierpinski triangle: after step 2, small shaded triangles are left.
Answer:
3.Drawing the Sierpinski triangle: after step 3, small shaded triangles are left.
Answer:
4.After step 1 of the Sierpinski triangle, the shaded area is of the first triangle's.
Fill in each blank
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5.Drawing the Sierpinski triangle from step 1 to step 2, triangles are taken out in all.
Answer:
6.After step 4 of the Sierpinski triangle, the shaded area is of the first triangle's.
Fill in each blank
Answer:
7.Drawing the Sierpinski triangle from step 1 to step 5, triangles are taken out in all.
Answer:
8.An equilateral triangle has an area of 128 cm². Made into a Sierpinski triangle, after step 3 the shaded area =
Answer:
9.An equilateral triangle has sides of 12 cm. Made into a Sierpinski triangle, after step 2 the perimeters of all the small shaded triangles add up to
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10.An equilateral triangle has an area of 192 cm². Made into a Sierpinski triangle, after step 3 the shaded area =
Answer:
11.An equilateral triangle has sides of 6 cm. Made into a Sierpinski triangle, after step 1 the perimeters of all the small shaded triangles add up to
Answer:

12.After step 1 of the Sierpinski triangle, the shaded area is less than half the first triangle's.
True or false?
Answer:

13.An equilateral card of 4 cm² is cut out as a Sierpinski triangle to step 1. The area left = cm², and the area cut away = cm².
Fill in each blank
Working:
Answer:
14.A wall hanging is a Sierpinski triangle drawn to step 3, the first triangle's area 256 cm². There are small shaded triangles, and the shaded area = cm².
Fill in each blank
Working:
Answer:
15.A craftsman makes a Sierpinski triangle in gold leaf to step 2, one sheet on each small shaded triangle at $7 a sheet. How much is the gold leaf?
Working:
Answer:
16.Drawing a Sierpinski triangle, at least how many steps make the shaded area less than 50% of the first triangle's?
Working:
Answer:
17.A wall hanging is a Sierpinski triangle drawn to step 4, the first triangle's area 256 cm². There are small shaded triangles, and the shaded area = cm².
Fill in each blank
Working:
Answer:
18.An equilateral card of 80 cm² is cut out as a Sierpinski triangle to step 2. The area left = cm², and the area cut away = cm².
Fill in each blank
Working:
Answer:

Show the answers and working

The Sierpinski triangle worksheet: Answers and working

1. 81
The picture has 3 shaded triangles; each step turns every one into 3: 3 × 3³ = 81.
2. 9
Times 3 each step: 3² = 9.
3. 27
Times 3 each step: 3³ = 27.
4. 3, 4
Each step keeps 34: (34)¹ = 34.
5. 4
Step k takes out one from each triangle left before it: 1 + 3 + 9 + … + 3^(2 − 1) = (3² − 1) ÷ 2 = 4.
6. 81, 256
Each step keeps 34: (34)⁴ = 81256.
7. 121
Step k takes out one from each triangle left before it: 1 + 3 + 9 + … + 3^(5 − 1) = (3⁵ − 1) ÷ 2 = 121.
8. 54 cm²
128 × (34)³ = 54 cm²; or: each small triangle is 128 ÷ 4³ = 2 cm², and there are 3³ of them.
9. 81 cm
Each small triangle has sides 12 ÷ 2² = 3 cm and a perimeter of 3 × 3 cm, and there are 3²: 81 cm. Or times 32 each step.
10. 81 cm²
192 × (34)³ = 81 cm²; or: each small triangle is 192 ÷ 4³ = 3 cm², and there are 3³ of them.
11. 27 cm
Each small triangle has sides 6 ÷ 2¹ = 3 cm and a perimeter of 3 × 3 cm, and there are 3¹: 27 cm. Or times 32 each step.

12. False
What's left is (34)¹: 34 and 916 are both over a half; 2764 is the first under it.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 3, 1
Left: 4 × (34)¹ = 3 cm²; cut away: 4 − 3 = 1 cm².
14. 27, 108
3³ = 27 triangles; each is 256 ÷ 4³ = 4 cm², so 108 cm² in all.
15. 63 dollars
3² small triangles at $7 each: 63 dollars.
16. 3
Times 34 each step: 75%, 56.25%, 42.19%, 31.64%, 23.73%, 17.80%, 13.35%, 10.01%, 7.51%, …; the first under 50% is step 3.
17. 81, 81
3⁴ = 81 triangles; each is 256 ÷ 4⁴ = 1 cm², so 81 cm² in all.
18. 45, 35
Left: 80 × (34)² = 45 cm²; cut away: 80 − 45 = 35 cm².

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