Important Questions Class 8 Maths Ganita Prakash – II Chapter 4 Exploring Some Geometric Themes

Short Questions

Q1. What is a fractal?

Answer: A fractal is a shape that shows the same pattern repeatedly at smaller scales.

Q2. What is the value of 

$R_n$

​ in Sierpinski Carpet?

Solution:
Each step produces 8 times the previous squares:

$R_{n+1} = 8R_n$

​, so

$R_n = 8^n$

.

Q3. What is

$H_1$

in Sierpinski Carpet?

Solution:

$H_1 = 1$

Initially, one central square is removed → 1 hole.

Q4. Name one example of fractal in nature.

Solution: Fern leaves show repeating patterns at smaller scales.

Q5. What is a net?

Solution: A net is a flat shape that can be folded to form a solid.

Example: Cube net unfolds into 6 squares.

Q6. How many faces does a cube have?

Solution:

Cube has 6 square surfaces.

Q7. What is a prism?

Solution: A prism has two identical polygon faces connected by parallelogram faces.

Example: Triangular prism.

Q8. What is a pyramid?

Solution: A pyramid has a polygon base and triangular faces meeting at a point.

Example: Square pyramid.

Q9. What is projection?

Solution: Projection is the image of an object on a plane.

Like a shadow or front/top view.

Q10. What is isometric projection?

Solution: A projection where all edges appear equal in length.

Example: Cube appears like a hexagon.

Long Questions

Q1. Explain the Sierpinski Carpet and find formulas for squares and holes.

Solution: The Sierpinski Carpet is made by dividing a square into 9 parts and removing the center repeatedly.

• Step 0: 1 square
• Step 1: 8 squares
• Step 2:
  $8^2 = 64$ 

  
  

So,

$$R_n=8^n$$

For holes:

$$H_{n+1}=H_n+R_{\mathrm{n<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}$$

Example:

• 
  $H_1 = 1$ 
• 
  $H_2=1+8=\mathrm{9<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"> </span>}$ 

Q2. Explain Sierpinski Triangle and its construction.

Solution: It is formed by dividing an equilateral triangle into 4 smaller triangles and removing the middle one.

Steps:

1. Start with triangle
2. Join midpoints
3. Remove central triangle
4. Repeat process

Each step reduces area and increases holes.

Q3. Explain Koch Snowflake construction and pattern.

Solution: It is formed by modifying each side of a triangle repeatedly.

Steps:

1. Divide each side into 3 parts
2. Add triangle on middle part
3. Remove middle segment

Pattern:

• Number of sides increases
• Shape becomes more complex

Q4. Explain nets of solids with examples.

Solution: A net is a 2D figure that folds into a 3D shape.

Examples:

• Cube → 6 squares
• Cylinder → rectangle + 2 circles
• Cone → sector of circle

Use of nets:

• Helps in construction
• Helps find surface area

Q5. Explain shortest path on a cuboid using nets.

Solution: Shortest path on a cuboid is found by unfolding it into a net.

Steps:

1. Open cuboid into flat net
2. Draw straight line between points
3. Measure length

Reason:

Shortest path on plane is a straight line, so same applies to unfolded surface.