NCERT solutions for Class 8 Mathematics Chapter Exploring Some Geometric Themes – complete answers & explanations

This blog provides clear and reliable NCERT solutions for Class 8 Mathematics Chapter Exploring Some Geometric Themes from Ganit Prakash II. In this chapter, students explore fascinating ideas like fractals, patterns that repeat at smaller scales, and ways to visualise three-dimensional shapes. These concepts help build strong spatial understanding and logical thinking, which are very useful in higher mathematics. Students also learn how shapes grow step by step, how nets form solids, and how to understand different views of objects. This blog is designed to help students understand the correct approach to answering questions and build confidence in exams. Download the worksheet and practice alongside solutions for better clarity. If your child needs extra support in understanding these concepts, Book a free trial now to get expert guidance.

What this NCERT chapter covers?

1. Understanding fractals and self-similar patterns such as the Sierpinski triangle and carpet.
2. Learning how shapes repeat and grow step by step using geometric rules.
3. Exploring patterns in number of shapes, holes, and areas in fractals.
4. Studying the Koch snowflake and how its sides and perimeter increase.
5. Visualising shapes mentally without drawing them on paper.
6. Identifying different profiles of solids from various viewpoints.
7. Understanding faces, edges, and vertices of 3D shapes.
8. Learning about nets of solids like cubes, prisms, pyramids, cylinders, and cones.
9. Exploring how solids can be constructed from flat surfaces.
10. Understanding shortest paths on surfaces using nets.
11. Learning projections and representation of solids on a plane.
12. Developing spatial reasoning and problem-solving skills useful for exams.

How to use these NCERT solutions?

1. First, read each question carefully and try solving it on your own.
2. Attempt drawing-based and activity questions practically wherever possible.
3. Use the provided answers to check your approach and accuracy.
4. Compare each step with the solution to understand the correct method.
5. Follow the same order of questions as given in your worksheet.
6. Revise mistakes and reattempt difficult questions for better understanding.
7. Parents and teachers can guide students by discussing concepts step-by-step.
8. Use diagrams and visualisation techniques to improve clarity.
9. Practice regularly to strengthen geometry concepts.
10. Focus on understanding patterns rather than memorising answers.

Important tips & tricks for students

1. Always draw neat diagrams for geometry questions.
2. Pay attention to patterns in fractals and sequences.
3. Understand how shapes change at each step instead of memorising.
4. Practice visualising shapes mentally to improve spatial skills.
5. Learn how nets fold into solids for better clarity.
6. Avoid skipping steps while solving pattern-based questions.
7. Carefully read activity-based instructions before attempting.
8. Use correct formulas where required and apply them properly.
9. Double-check answers involving counting shapes or sides.
10. Stay consistent with practice to improve accuracy and speed.

NCERT solutions – complete answer key

Figure it Out 
(Page 72)

Explanation:
Draw an equilateral triangle. Divide it into 4 equal
smaller triangles and remove the central triangle. Repeat
for remaining triangles till Step 2.

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)

Explanation:
Start with an equilateral triangle and follow steps to
form snowflake till Step 2.

Perimeter = 3 × (4/3)ⁿ

Answer:
Shape = Hexagon

Number of sides = 3 × 4ⁿ

Answer:
Shape left = Octagon 
Reassemble corners to form a square.

Visualisation Activities
(Page 75–76)

Explanation:
Write your name and read letters in reverse visually.

Answer:
Shape = Octagon

Example:
Cylinder

Example:
Triangular prism

Example:
Cylinder (top view)

Example:
Cone (front view)

Example:
Cube (front view)

Example:
Cone

Example:
Cone

Example:
Pentagonal prism

Figure it Out 
(Page 80–81)

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

Explanation:
There are 11 distinct nets of a cube (draw all possible).

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

Math Talk / Activities 
(Page 81–83)

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

Math Talk
(Page 82)

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

Cone:
Sector of a circle

Shortest Path 

(Page 84–87)

Answer:
Shortest path = straight line on suitable net

Given case:
d² = 24² + 32² 
d = 40 cm

Figure it Out
(Page 92)

Answer:
Projected lengths vary depending on orientation

Figure it Out 

(Page 93)

Answer:
Match each object with correct projections based on
shape alignment

Answer:
Cube → square 
Cuboid → rectangle 
Cylinder → rectangle/circle 
Cone → triangle/circle 
Prism → rectangle/triangle 
Pyramid → triangle

Figure it Out
(Page 95–96)

Explanation:
Draw projections as per cube arrangement

(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 

Explanation:
Construct solids using identical cubes

Figure it Out
(Page 97)

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

Figure it Out 
(Page 100)

Answer:
No additional distinct shapes beyond
rotations/reflections

Answer:
Square, hexagon, rectangle (depending on orientation)

Figure it Out 
(Page 101–102)

Explanation:
Draw figures on isometric grid carefully following axes

Answer:
Path appears impossible due to perspective illusion

Figure it Out
(Page 101–102)

(i) Not possible using real cubes 
Front, Top, Side views differ 

(ii) Draw on isometric grid 

(iii) Illusion works due to inconsistent perspective

Activities / Observations

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

Why NCERT solutions help students?

NCERT solutions help students build strong concepts, improve problem-solving skills, and prepare effectively for exams. They provide a clear understanding of the correct answering approach and boost confidence by ensuring students practice with accurate and reliable answers aligned with NCERT expectations.

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