Question 1

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.

Accepted Answer

Step 0: Take cut out of an equilateral triangle Δ.
Step 1: Divide it into 4 equilateral triangles by joining the midpoints of each of the sides.
Remove the central triangle.
Step 2: Divide each of the remaining 3 equilateral triangles into four equilateral triangles and remove the central triangle in each of them.
Step 3: Repeat the steps again and again to get Sierpinski’s gasket.

Question 2

Find the number of holes and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.

Accepted Answer

Number of holes in
Step 0: 0 hole
Step 1: 1 hole
Step 2: 1 + 3 = 4 holes
Step 3: 1 + 3 + 9 = 13 holes
Step 4: 1 + 3 + 9 + 27 = 40 holes
.
.
.
Step n: $3^{1-1}+3^{2-1}+3^{3-1}+\ldots+3^{n-1}$ holes
$3^0+3^1+3^2+\ldots+3^{n-1}$ holes

Question 3

Find the area of the region remaining at the nth step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to the 1 sq. unit.

Accepted Answer

(a) Let the side of the square of Sierpinski’s Carpet be 1 square unit.
As one-ninth of the square is removed, and one-eighth remains in each step.
Step 0: Area of whole square = (1)
2
= 1
Step 1: Area of remaining region = $\frac{8}{9}(1)^2=\frac{8}{9}$
Step 2: Area of remaining region = $\frac{8}{9} \times \frac{8}{9}=\frac{64}{81}$
Step 3: Area of remaining region = $\frac{8}{9} \times \frac{64}{81}=\frac{8^3}{9^3}=\frac{512}{729}$
Area of remaining region after nth step = $\left(\frac{8}{9}\right)^n$ sq. units
(b) Let the area of the triangle of Sierpinski’s Gasket be 1 square unit.
One-fourth of the triangle is removed in each step, and three-fourth remains.
Step 0: Area of the whole triangle = 1 sq. unit.
Step 1: Area of remaining region = $\frac {3}{4}$ sq. units.
Step 2: Area of remaining region = $\frac{3}{4} \times \frac{3}{4}$ square units = $\frac {9}{16}$ sq. units
Area of remaining region after nth step = $\left(\frac{3}{4}\right)^n$ sq. units
Figure It Out (Page 73)

Question 4

Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left? (Page 76)

Accepted Answer

KLMN is a square.
Each side of the square is divided into three equal parts.
When the four corners of the square are cut off, we get an octagon.
Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can imagine the solid passing through a wall like Tom did, and leaving a hole of the appropriate shape. (Page 77)

Question 5

A solid whose profile has a square outline.

Accepted Answer

A solid with a profile that has a square outline is a CUBE.

Source: This question is from Exploring Some Geometric Themes, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Exploring Some Geometric Themes: Solid, Profile, Square, Exploring Some Geometric Themes, Mathematics. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Exploring Some Geometric ThemesQ1: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the...
Q12: A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint.
Q2: Find the number of holes and the triangles that remain at each step of the shape sequence that...
Q3: Find the area of the region remaining at the nth step in each of the shape sequences that lead to...
Q4: Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What...
Practice more questions from Exploring Some Geometric Themes &mdash; Mathematics, Class 8 CBSE

Question 6

A solid whose profile has a circular outline.

Accepted Answer

A solid whose profile has a circular outline is a sphere.

Source: This question is from Exploring Some Geometric Themes, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Exploring Some Geometric Themes: Solid, Profile, Circular, Exploring Some Geometric Themes, Mathematics. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Exploring Some Geometric ThemesQ1: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the...
Q12: A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint.
Q2: Find the number of holes and the triangles that remain at each step of the shape sequence that...
Q3: Find the area of the region remaining at the nth step in each of the shape sequences that lead to...
Q4: Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What...
Practice more questions from Exploring Some Geometric Themes &mdash; Mathematics, Class 8 CBSE

Question 7

A solid whose profile has a triangular outline.

Accepted Answer

A solid whose profile has a triangular outline is a triangular pyramid.
As we saw with the elephant, a given solid might have very different profiles from different viewpoints. Can you visualise solids that have the following contrasting profiles?
Spend some time on this, and if you are finding it difficult to visualise, you may look around and use objects that are around you, or that you will make in the next section. Feel free to consider viewpoints from any direction, including directly above the object. (Page 77)

Question 8

A solid with a trapezium-shaped profile from one viewpoint and a circular one from another viewpoint.

Accepted Answer

A truncated cone or a cone cut off from a larger cone.
This shape is called a frustum.

Source: This question is from Exploring Some Geometric Themes, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Exploring Some Geometric Themes: Solid, Trapezium, Shaped, Profile, Viewpoint, Circular. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Exploring Some Geometric ThemesQ1: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the...
Q12: A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint.
Q2: Find the number of holes and the triangles that remain at each step of the shape sequence that...
Q3: Find the area of the region remaining at the nth step in each of the shape sequences that lead to...
Q4: Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What...
Practice more questions from Exploring Some Geometric Themes &mdash; Mathematics, Class 8 CBSE

Question 9

A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint.

Accepted Answer

Pentagon base prism.
4.3 Making Solids, 4.4 Special Solids
Intext Questions (Pages 79-80)

Source: This question is from Exploring Some Geometric Themes, Mathematics &mdash; Class 8, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Exploring Some Geometric Themes: Solid, Pentagonal, Profile, Viewpoint, Rectangular, Another. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 8 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Exploring Some Geometric ThemesQ1: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the...
Q2: Find the number of holes and the triangles that remain at each step of the shape sequence that...
Q3: Find the area of the region remaining at the nth step in each of the shape sequences that lead to...
Q4: Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What...
Q5: A solid whose profile has a square outline.
Practice more questions from Exploring Some Geometric Themes &mdash; Mathematics, Class 8 CBSE