Maths ·Class 9 Maths

Observe the Sierpiriski triangle and try to answer the following questions a. How many black triangles Class 9

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· Jul 10, 2026 · Reviewed & updated Sep 17, 2026 · 1 min read

Observe the Sierpiriski triangle and try to answer the following questions a. How many black triangles Class 9

Question 1.

Observe the Sierpiriski triangle and try to answer the following questions Class 9

a. How many black triangles are there in Stages 0 to 3 of above figure showing Sierpiriski triangle?

b. Can you predict the number of black triangles at Stages 4 and 5?

c. Can you find a rule for the number of black triangles at the nth stage?

d. Suppose the area of the triangle (that is, the black region) in Stage 0 is 1 square unit. What is the area of the black region in Stages 1, 2 and 3? What will be the area of the black region in Stages 4 and 5? Find a rule for the area of the black region at the n^th stage. What happens to this area as n, the number of stages, goes on increasing?

Answer:

Stage (n)	0	1	2	3
Number of black triangles (t_n)	1	3	9	27

b. Each stage is 3 times the previous one

∴ Stage 4 will have, 27 × 3 = 81 black triangles and

Stage 5 will have, 81 × 3 = 243 black triangles

c. Rule for n^th stage:

Number of triangles

= 3 × 3 × 3 × .... (n times).

∴ Number of triangles for n^th stage = 3ⁿ.

d. Given, In Stage 0,

Area of black region = 1 unit.

Each stage keeps $\frac{3}{4}$ part of the previous area

Stage (n)	1	2	3
Area of black region	$\frac{3}{4}$	$\left(\frac{3}{4}\right)^2$	$\left(\frac{3}{4}\right)^3$

∴ Area of black region region at 4^th stage

$=\left(\frac{3}{4}\right)^4$

∴ Area of black region region at 5^th stage

$=\left(\frac{3}{4}\right)^5$

∴ Area of black region at n^th stage = $\left(\frac{3}{4}\right)^n$

∴ As number of stages n increases, Area becomes smaller and smaller and it gets closer to 0.