NCERT Solutions for Class 8 Maths Chapter A Square and a Cube

This worksheet for Class 8 Maths Chapter A Square and a Cube helps students understand square numbers, cube numbers, and their properties in a clear and structured way. It focuses on identifying patterns, solving numerical problems, and applying mathematical reasoning. This worksheet provides complete and accurate NCERT Solutions, making it a reliable resource for practice, revision, and concept clarity.

Chapter summary: stories, poems & themes

This chapter is concept-based and focuses on number patterns and mathematical relationships. It includes puzzle-based learning, observation activities, and pattern recognition exercises. Students explore square numbers, cube numbers, and their properties through logical thinking and problem-solving tasks. The chapter is largely activity-based and encourages analytical reasoning.

What this NCERT chapter covers?

• Understanding square numbers and perfect squares  
• Identifying patterns in squares and cube numbers  
• Learning properties of numbers such as parity and factors  
• Finding square roots and cube roots  
• Solving problems using prime factorisation  
• Observing number patterns and relationships  
• Applying concepts in real-life situations  

How to use these NCERT solutions?

Students should first try solving each question on their own and then use these NCERT Solutions to check their answers. Parents and teachers can guide students by explaining the steps and concepts used in each solution. The solutions follow the exact order and structure of the NCERT worksheet, which helps in systematic learning and easy revision.

Student tips & learning tricks

Students should remember that perfect squares follow specific patterns in their unit digits. While solving problems, carefully check whether numbers can be grouped into pairs for identifying perfect squares. Practice estimating square roots using nearby perfect squares. Pay attention to patterns like differences between consecutive squares and properties of cubes to avoid common mistakes.

Why NCERT solutions are important?

NCERT Solutions help students build strong mathematical concepts and improve problem-solving skills. They ensure that students follow the correct methods as per NCERT guidelines. Regular practice with accurate solutions increases confidence and helps students perform better in school assessments and exams.

Complete answer key – NCERT solutions

Puzzle Section

1. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100  
Explanation: Lockers remain open if they are toggled an odd number of times. Only perfect squares have an odd number of factors, so square-numbered lockers stay open.

2. 2, 3, 5, 7, 11  
Explanation: Prime numbers have exactly two factors (1 and the number itself), so their lockers are toggled twice.

1.1 Square Numbers

1. 12, 23, 37, 58, 77  
Explanation: Squares never end with 2, 3, 7, or 8.

2. (ii) 342, (iv) 562, (vi) 822  
Explanation: These numbers end with 6, matching the condition.

3. 6 zeros  
Explanation: Squaring doubles the number of trailing zeros.

4. If a number is even, its square is even. If a number is odd, its square is odd.

5. 36² = 1296  
Explanation: The difference between consecutive squares is the next odd number (2n+1).

6. 2n – 1  
Explanation: Odd numbers follow the formula 2n – 1.

7. Between n² and (n+1)², there are 2n numbers.

8. 1–100 → 10 squares  
  101–200 → 4 squares  
  Largest square < 1000 → 961 (31²)

9. 1 + 3 = 4 = 2²  
  3 + 6 = 9 = 3²  
  6 + 10 = 16 = 4²  
  10 + 15 = 25 = 5²

10. √49 = 7

11. ±8  
Explanation: Both +8 and -8 squared give 64.

13. Yes, √324 = 18  
Explanation: Prime factorisation groups evenly.

14. No  
Explanation: Prime factors cannot be paired completely.

15. 1156 → Perfect square, √1156 = 34  
   2800 → Not a perfect square

16. √1936 = 44  
Explanation: 1936 lies between 40² and 45², closer to 44².

17. √250 ≈ 15.8 (closer to 16)  
Explanation: 250 lies between 15² and 16².

18. Side length = 11 cm  
Explanation: 11² = 121 < 125, but 12² = 144 > 125.

Figure it Out (Squares)

1. 2032, 2048, 1027  

2. 108² ends with 4  

3. 126² = 15876  

4. Side length = 21 m (√441)  

5. 900  

6. √197568 = 444  

7. (i) 33 numbers  
  (ii) 199 numbers  

8. 212, 152, 192  

9. 256 tiny squares = 2⁸  

1.2 Cubic Numbers

1. 27  
Explanation: 3³ = 27.

2. 12³ = 1728  
  15³ = 3375  
  6³ = 216  
  16³ = 4096  
  7³ = 343  
  8³ = 512  
  9³ = 729  
  10³ = 1000  
  20³ = 8000  

3. 0, 1, 4, 5, 6, 9  
Explanation: Observed from cube patterns.

4. No  
Explanation: Cubes of multiples of 10 end with three or more zeros.

5. ∛64 = 4  
  ∛512 = 8  
  ∛729 = 9  

Figure it Out (Cubes)

1. ∛27000 = 30, ∛10648 = 22  

2. Multiply by 3  

3. (i) False  
  (ii) False  
  (iii) True  
  (iv) True  
  (v) False  

4. 1331 → 11  
  4913 → 17  
  12167 → 23  
  32768 → 32  

5. 67³ – 66³ = 13411  

Puzzle Time

Square Pairs Arrangement:
16 – 9 – 7 – 2 – 14 – 11 – 5 – 20 – 4 – 21 – 15 – 10 – 6 – 3 – 13 – 12 – 1  

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