NCERT Solutions for Class 8 Maths Chapter 4 Exploring Some Geometric Themes

This worksheet helps students understand Class 8 Mathematics NCERT Chapter Exploring Some Geometric Themes in a clear and structured way. This chapter focuses on interesting concepts like fractals, nets of solids, projections, and visualisation of 3D shapes. These topics are important because they build strong spatial understanding and connect mathematics with real-world objects and patterns. This worksheet provides complete and accurate NCERT Solutions based strictly on the chapter, helping students practice and verify their answers confidently.

Chapter summary: stories, poems & themes

This chapter introduces geometric themes through activity-based and visual learning. Students explore fractals such as the Sierpinski Triangle, Sierpinski Carpet, and Koch Snowflake, which show repeating patterns at smaller scales. The chapter also focuses on visualising solids, understanding nets, and observing projections from different views. It is largely activity-based, involving drawing, observation, imagination, and hands-on tasks to develop spatial thinking.

What this NCERT chapter covers?

• Understanding fractals and self-similar patterns 
• Visualising geometric shapes and solids 
• Learning about nets of 3D shapes like cubes, prisms, and pyramids 
• Exploring projections such as front view, top view, and side view 
• Developing observation and drawing skills using isometric grids 
• Understanding shortest paths on surfaces using nets 

How to use these NCERT solutions?

Students should first attempt all questions on their own before checking the answers in this worksheet. This helps improve problem-solving skills and understanding. Parents and teachers can use these solutions to guide students and ensure correct learning. The answers follow the exact NCERT order and structure, making it easier for revision and exam preparation.

Student tips & learning tricks

• Carefully follow each step in drawing-based questions 
• Practice visualisation instead of memorising answers 
• Pay attention to patterns in fractals and sequences 
• Avoid skipping steps in geometry constructions 
• Use neat diagrams wherever required 
• Understand the concept of nets and projections clearly before solving 

Why NCERT solutions are important?

NCERT Solutions help students build strong conceptual understanding based on the official curriculum. They ensure that answers are accurate and aligned with exam expectations. These solutions improve confidence, support regular practice, and help students prepare effectively for school assessments and exams.

Complete answer key – NCERT solutions

Figure it Out (Page 72)

1. 
Step 0: 
Triangles = 1, Holes = 0 

Step 1: 
Triangles = 3, Holes = 1 

Step 2: 
Triangles = 9, Holes = 1 + 3 = 4 

Step 3: 
Triangles = 27, Holes = 1 + 3 + 9 = 13 

General: 
Triangles = 3ⁿ 
Holes = 1 + 3 + 3² + ... + 3ⁿ⁻¹ 

Sierpinski Carpet: 
Remaining area = (8/9)ⁿ 

Sierpinski Triangle: 
Remaining area = (3/4)ⁿ 

Figure it Out (Page 73)

1. 
Perimeter = 3 × (4/3)ⁿ 

2. 
Shape = Hexagon 
Number of sides = 3 × 4ⁿ 

3. 
Shape left = Octagon 
Reassemble corners to form a square 

Visualisation Activities (Page 75–76)

1. 
student-generated activity 

2. 
Shape = Octagon 

3. 
student-generated activity 

4. 
student-generated activity 

5–12. 
Cylinder 
Triangular prism 
Cylinder (top view) 
Cone (front view) 
Cube (front view) 
Cone 
Cone 
Pentagonal prism 

Figure it Out (Page 80–81)

1. 
(i) Yes 
(ii) No 
(iii) Yes 
(iv) No 
(v) Yes 
(vi) No 

2. 
There are 11 distinct nets of a cube 

3. 
Draw rectangular faces with given dimensions and arrange to form net 

Math Talk / Activities (Page 81–83)

Draw nets of: 
Regular tetrahedron (2 possible nets) 
Square pyramid 
Triangular prism 

Math Talk (Page 82)

Rectangle dimensions: 
Length = circumference of base 
Width = height of cylinder 

Cone: 
Sector of a circle 

Shortest Path (Page 84–87)

Shortest path = straight line on suitable net 

Given case: 
d² = 24² + 32² 
d = 40 cm 

Figure it Out (Page 92)

Projected lengths vary depending on orientation 

Figure it Out (Page 93)

Match each object with projections: 
Cube → square 
Cuboid → rectangle 
Cylinder → rectangle/circle 
Cone → triangle/circle 
Prism → rectangle/triangle 
Pyramid → triangle 

Figure it Out (Page 95–96)

(i) Cube 
(ii) Cylinder 
(iii) Cone 
(iv) Prism 
(v) Pyramid 
(vi) Cuboid 
(vii) Sphere 

(i) Side view: depends on arrangement 
Top view: depends on arrangement 

(ii) Add cubes accordingly 

(iii) Add more cubes accordingly 

(iv) Various combinations possible 

Figure it Out (Page 97)

Count cubes layer-wise (exact number depends on figure) 

Figure it Out (Page 100)

No additional distinct shapes beyond rotations/reflections 

Square, hexagon, rectangle (depending on orientation) 

Figure it Out (Page 101–102)

1. 
Not possible using real cubes 
Front, Top, Side views differ 

2. 
Draw on isometric grid 

3. 
Illusion works due to inconsistent perspective 

Activities / Observations

Perform all drawing, folding, and observation-based tasks practically as instructed in the book. 

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