Cut out the middle

The idea

The Sierpiński triangle is made by joining the midpoints of an equilateral triangle's sides, removing the central triangle and repeating; at stage n it has 3ⁿ black triangles of total area (3/4)ⁿ, both GPs.

Cut and repeat

Take a paper triangle. Join the midpoints of its three sides. That makes 4 small triangles. Remove the one in the middle. Then do the same to each of the 3 triangles left, and again, and again …

How many black triangles are there after cutting 3 times (Stage 3)?

• 9
• 12
• 27

Make the Sierpiński triangle

Press the button to cut. Watch the table: how many black triangles are there, and how much black area is left?

Two GPs

The Sierpiński triangle is made by joining the midpoints of an equilateral triangle's sides, removing the central triangle and repeating. At stage n it has 3ⁿ black triangles of total area (3⁄4)ⁿ, both GPs.

Each black triangle is replaced by 3 smaller ones, so the count is multiplied by 3 each stage: 1, 3, 9, 27, 81, 243, … (r = 3).

Each cut leaves 3/4 of the black area, so the area is multiplied by 3/4 each stage: the common ratio is 3/4. If stage 0 has area 1, stage n has area (3⁄4)ⁿ.

The number of triangles grows very quickly, while the black area gets closer and closer to 0.

Notes

The Sierpiński triangle: join the midpoints of an equilateral triangle's sides, remove the central triangle and repeat. At stage n there are 3ⁿ black triangles of total area (3⁄4)ⁿ, both GPs.

Check yourself

How many black triangles are there at Stage 6? (Stage 5 has 243.)

Answer: 729

243 × 3 = 729. That is 3⁶.

What is the black area at Stage 2, if the whole triangle at Stage 0 has area 1?

As the stage number grows, what happens?

By what number is the count of black triangles multiplied at each stage?

Answer: 3

Cutting out the middle leaves 3 small black triangles for each black triangle, so the count is multiplied by 3. The common ratio is 3.

• 1/2. The area is not reduced by 1/4 twice from 1. Each stage keeps 3/4 of the area before: 3/4 × 3/4.
• 9/16 — correct. Yes. Stage 1 keeps 3/4 of the area, and Stage 2 keeps 3/4 of that again. So 3/4 × 3/4 gives 9/16 of the original area.
• 3/4. 3/4 is the area at Stage 1. Stage 2 keeps 3/4 of it again.
• The number of black triangles grows very fast, while the black area shrinks towards 0 — correct. Yes. The count is multiplied by 3 each stage, and the area by 3/4.
• The number of black triangles and the black area both grow. Only the number of triangles grows. The black area keeps being multiplied by 3/4, so it shrinks.
• Both shrink. The number of triangles is multiplied by 3 at every stage, so it grows.