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CBSE Class 8 Mathematics Chapter 5 Squares and Square Roots Worksheet with Answers

Chapter 5 Squares and Square Roots is a foundational topic in CBSE Class 8 Mathematics that builds number sense and computational skills critical for higher algebra and geometry. This printable worksheet offers systematic practice across all NCERT topics including properties of square numbers, Pythagorean triplets, and square root extraction through prime factorisation and long division methods. Designed for 90-minute timed practice at medium-to-hard difficulty, it serves both classroom assessment and home revision needs.

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Key takeaways

  • ✓Worksheet contains 30+ questions covering all topics from NCERT Class 8 Mathematics Chapter 5 on Squares and Square Roots
  • ✓Structured in five graded sections: MCQs, fill-in-the-blanks, matching, short answers, and HOTS long-answer questions
  • ✓Includes one case-study question following the latest CBSE examination pattern for 2025
  • ✓Complete answer key provided with brief explanations and working steps for every question
  • ✓Recommended time: 90 minutes; Difficulty level: Medium to Hard, suitable for term-end revision
  • ✓Covers properties of square numbers, Pythagorean triplets, prime factorisation method, and long division method for square roots
  • ✓Ideal for self-assessment before CBSE periodic tests and annual examinations

Quick Chapter Recap: Squares and Square Roots

Before attempting the worksheet, revisit these core concepts from NCERT Class 8 Mathematics Chapter 5. A square number is obtained when a number is multiplied by itself; for example, 49 is the square of 7. Square numbers have unique properties: they end only in 0, 1, 4, 5, 6, or 9; a number ending in 2, 3, 7, or 8 can never be a perfect square. The number of zeros in a perfect square is always even. Pythagorean triplets are sets of three natural numbers (a, b, c) where a² + b² = c², such as (3, 4, 5) or (5, 12, 13). To find square roots, two principal methods are taught: prime factorisation, where you pair prime factors and take one from each pair, and the long division method, which works for both perfect and non-perfect squares and is especially useful for large numbers. Understanding these concepts thoroughly ensures accuracy in solving numerical problems and lays groundwork for Pythagoras theorem in geometry.
  • Perfect squares end in 0, 1, 4, 5, 6, or 9 only
  • Square of an even number is even; square of an odd number is odd
  • Between consecutive squares n² and (n+1)², there are 2n non-square numbers
  • Pythagorean triplet: For any m > 1, numbers 2m, m²−1, m²+1 form a triplet
  • Prime factorisation method pairs prime factors to extract square root
  • Long division method works for decimals and non-perfect squares

Section A: Multiple Choice Questions (MCQs)

This section contains six multiple-choice questions testing your conceptual clarity and quick problem-solving skills. Each question carries one mark. Choose the correct option and write the letter (a, b, c, or d) in your answer sheet. These MCQs cover properties of squares, identification of perfect squares, Pythagorean triplets, and basic square root calculations. Remember that perfect squares have an odd number of total factors, and the units digit pattern repeats predictably. Time allocation for this section should be around 10-12 minutes. Read each question carefully, eliminate obviously wrong options first, and then calculate or reason through the remaining choices. This pattern of MCQs mirrors the objective questions asked in CBSE periodic tests and reflects the style seen in recent Class 8 Mathematics annual examinations.
  • Q1. Which of the following numbers is NOT a perfect square? (a) 256 (b) 361 (c) 444 (d) 529
  • Q2. The square of 58 will have the units digit: (a) 2 (b) 4 (c) 6 (d) 8
  • Q3. How many natural numbers lie between 18² and 19²? (a) 36 (b) 37 (c) 38 (d) 18
  • Q4. Which of these is a Pythagorean triplet? (a) (6, 7, 8) (b) (8, 15, 17) (c) (9, 12, 16) (d) (10, 11, 12)
  • Q5. The smallest number by which 2700 must be multiplied to make it a perfect square is: (a) 2 (b) 3 (c) 5 (d) 6
  • Q6. The square root of 0.0169 is: (a) 0.13 (b) 0.013 (c) 1.3 (d) 13

Section B: Fill in the Blanks

Complete the following five statements by filling in the correct word, number, or mathematical expression in the blank spaces provided. Each blank carries one mark. This section tests your recall of definitions, properties, and formulae from NCERT Class 8 Mathematics Chapter 5. Write your answers precisely; numerical answers should be exact unless otherwise specified. Allocate approximately 8-10 minutes for this section. These questions focus on pattern recognition in square numbers, properties of digits, and the relationship between squares and square roots. Ensure you have memorised the squares of numbers from 1 to 25, as they form the foundation for quick mental calculations.
  • Q7. A number having 2, 3, 7, or 8 at the units place is never a __________ square.
  • Q8. The square root of 14641 is __________.
  • Q9. If m = 5, the Pythagorean triplet using the formula (2m, m²−1, m²+1) is __________.
  • Q10. The smallest perfect square divisible by 3, 4, 5, and 6 is __________.
  • Q11. There are __________ natural numbers between n² and (n+1)².

Section C: True or False Statements

Determine whether each of the following five statements is True or False. Write 'T' for true and 'F' for false in your answer sheet. Each correct answer carries one mark. This section assesses your grasp of fundamental properties and common misconceptions related to squares and square roots. Allocate around 8 minutes for this section. If a statement is false, you should be able to provide a counter-example or corrected version during self-review. Pay close attention to words like 'always,' 'never,' and 'only,' which often determine the truth value of mathematical statements. Understanding these properties deeply will help in both objective and subjective examinations conducted by CBSE.
  • Q12. The sum of two perfect squares is always a perfect square.
  • Q13. A perfect square can have an odd number of zeros at the end.
  • Q14. Every Pythagorean triplet consists of one even and two odd numbers.
  • Q15. The square root of a perfect square of n digits will have n/2 digits (if n is even).
  • Q16. The square of a prime number has exactly three factors.

Section D: Short Answer Questions (2-3 Marks Each)

Answer the following five questions in brief, showing necessary working steps. Each question carries 2 or 3 marks as indicated. Write complete solutions; marks are awarded for method as well as the final answer. Time allocation: 25-30 minutes. These questions require you to apply the methods learned in NCERT Class 8 Mathematics Chapter 5, including finding square roots by prime factorisation, determining the smallest multiplier or divisor to make a number a perfect square, and working with Pythagorean triplets. Show all steps clearly. When using prime factorisation, write the factor tree or division method neatly. For long division method, draw the standard division layout with proper placement of digits. Partial credit is often given in CBSE marking schemes for correct method even if the final answer has a minor error.
  • Q17. Find the square root of 3969 using the prime factorisation method. (2 marks)
  • Q18. What is the smallest number by which 1575 should be divided to get a perfect square? Also find the square root of the resulting number. (3 marks)
  • Q19. Find the Pythagorean triplet whose smallest member is 12. (2 marks)
  • Q20. A hall has area 5184 m². If it is a perfect square, find the side of the hall using the long division method. (3 marks)
  • Q21. Between which two consecutive natural numbers does √150 lie? (2 marks)

Section E: Long Answer and HOTS Questions (4-5 Marks Each)

Solve the following three higher-order thinking questions with complete working and justification. Each question carries 4 or 5 marks. Time allocation: 25-30 minutes. These questions require multi-step reasoning, application of multiple concepts, and often a synthesis of properties learned throughout the chapter. CBSE Class 8 Mathematics examinations increasingly include such HOTS (Higher Order Thinking Skills) questions to test analytical ability beyond rote learning. Write clearly, justify each step, and double-check your arithmetic. For questions involving finding unknowns, set up equations carefully. When proving properties or patterns, use algebraic expressions and logical reasoning. Neatness and structured presentation fetch additional marks under CBSE marking guidelines.
  • Q22. A gardener wants to plant trees in a square garden such that the number of rows equals the number of trees in each row. If he has 2000 trees, how many more trees does he need to complete the perfect square arrangement? (4 marks)
  • Q23. Prove that the difference between the squares of two consecutive natural numbers is equal to their sum. Use this property to find the value of 101² − 100² without actual multiplication. (5 marks)
  • Q24. A triplet of numbers is (8, 15, 17). Verify whether it is a Pythagorean triplet. Also, find the perimeter and area of the right-angled triangle formed by these numbers as sides. (4 marks)

Case-Study Based Question (4 Marks)

Read the following case study carefully and answer the sub-questions that follow. This question pattern aligns with the latest CBSE examination format introduced for competency-based assessment. The case study integrates real-life context with mathematical concepts from Squares and Square Roots. Time allocation: 10-12 minutes. Marks are distributed across four sub-parts, each testing a different skill: comprehension, application, calculation, and inference. Answer all parts; even if you are unsure about one sub-question, attempt the others as they may be independent. Such case-study questions appeared in CBSE Class 10 board papers from 2021 onwards and are now being adapted for Class 8 internal assessments as well. They test your ability to extract relevant information, model problems mathematically, and apply Chapter 5 concepts in practical scenarios.

Answer Key: Section A (MCQs)

Below are the correct answers with brief explanations for all six multiple-choice questions in Section A. Cross-check your responses and understand the reasoning behind each correct option. For any question you answered incorrectly, revisit the corresponding NCERT Class 8 Mathematics Chapter 5 section and clarify the concept before moving on. These explanations are designed to reinforce learning, not just to provide the right letter choice. If you scored below four out of six, consider revising the properties of square numbers and Pythagorean triplets once more. Regular practice with such MCQs sharpens pattern recognition and builds speed for timed tests. Parents can use this answer key to guide home revision sessions, ensuring the child understands the 'why' behind each answer rather than memorising options blindly.
  • A1. (c) 444. Explanation: 444 = 2×2×3×37. Since 3 and 37 are unpaired, 444 is not a perfect square.
  • A2. (b) 4. Explanation: 58 ends in 8; 8² = 64, so units digit is 4. Alternatively, (58)² = (60−2)² ends in 4.
  • A3. (a) 36. Explanation: Between n² and (n+1)² there are 2n numbers. Here n=18, so 2×18 = 36.
  • A4. (b) (8, 15, 17). Explanation: 8² + 15² = 64 + 225 = 289 = 17². It is a valid Pythagorean triplet.
  • A5. (b) 3. Explanation: 2700 = 2²×3³×5². Unpaired prime is 3. Multiply by 3 to get 2²×3⁴×5², a perfect square.
  • A6. (a) 0.13. Explanation: √0.0169 = √(169/10000) = 13/100 = 0.13.

Answer Key: Sections B, C, and Case Study

Here are the correct answers for fill-in-the-blanks (Section B), true/false statements (Section C), and the case-study question. Each answer includes a brief explanation or working to help you verify your method and understand any mistakes. For fill-in-the-blanks, ensure your numerical answers match exactly; for formula-based answers, verify substitution steps. In true/false questions, if you marked an answer incorrectly, write down a correct counter-example or the proper reasoning in your revision notes. For the case study, marks are earned both for correct numerical answers and for showing method. If you lost marks in sub-parts (ii) or (iv), check your arithmetic and conceptual steps carefully. This section is crucial for self-assessment before CBSE periodic tests, as fill-ups and case studies together can account for 8-10 marks in a typical 40-mark worksheet or internal exam.
  • A7. perfect (A number ending in 2,3,7,8 cannot be a perfect square.)
  • A8. 121 (14641 = 11×11×11×11 = 11⁴, so √14641 = 11² = 121.)
  • A9. (10, 24, 26) (Using m=5: 2m=10, m²−1=24, m²+1=26.)
  • A10. 900 (LCM of 3,4,5,6 is 60. Smallest perfect square multiple of 60 is 900 = 30².)
  • A11. 2n (Standard property: between n² and (n+1)² lie 2n natural numbers.)
  • A12. False (Example: 2² + 3² = 4 + 9 = 13, not a perfect square.)
  • A13. False (Perfect squares have even number of zeros; odd zeros mean it is not a perfect square.)
  • A14. False (E.g., triplet (3,4,5): 3 and 5 odd, 4 even. But (8,15,17): 8 even, 15 and 17 odd. Pattern varies.)
  • A15. True (E.g., 4-digit square 1296 has 2-digit root 36; 6-digit square 104976 has 3-digit root 324.)
  • A16. True (If p is prime, p² has factors 1, p, p². Exactly three factors.)
  • Case Study A(i): 39 students per row (√1521 = 39.)
  • Case Study A(ii): 1521+80=1601. √1601≈40.01. Next perfect square is 1681=41². Need 1681−1601=80 more students.
  • Case Study A(iii): Perimeter = 4×39 = 156 m.
  • Case Study A(iv): 1521÷39 = 39 rows.

Answer Key: Section D (Short Answer Questions)

This section provides step-by-step solutions for all five short-answer questions in Section D. Each solution demonstrates the standard CBSE marking scheme approach, showing method marks and final answer marks separately. When you compare your work, check not only the final answer but also the sequence of steps. For prime factorisation questions, ensure you have written factors in pairs and extracted roots correctly. For questions involving finding the smallest multiplier or divisor, verify that your prime factor grouping is accurate. If your answer differed, identify whether the error was conceptual (wrong method), computational (arithmetic mistake), or presentational (incomplete working). CBSE examiners award partial marks generously if the method is sound, so always show full working even under time pressure. Use these solutions as model answers when practicing similar questions from NCERT exercise 5.3 and 5.4.
  • A17. 3969 = 3×1323 = 3×3×441 = 3×3×21×21 = 3×3×3×7×3×7 = 3⁴×7². Pairing: (3²)²×7². √3969 = 3²×7 = 9×7 = 63. (2 marks: 1 for factorisation, 1 for final answer)
  • A18. 1575 = 3²×5²×7. Unpaired prime is 7. Divide by 7: 1575÷7 = 225. √225 = 15. (3 marks: 1 for factorisation, 1 for identifying divisor, 1 for square root)
  • A19. Smallest member 12. Use 2m=12 ⇒ m=6. Triplet: (2m, m²−1, m²+1) = (12, 35, 37). Verify: 12²+35²=144+1225=1369=37². (2 marks: 1 for finding m and triplet, 1 for verification)
  • A20. Area = 5184 m². Using long division for √5184: Group digits as 51|84. Largest n: n²≤51 ⇒ n=7 (49). Remainder 2. Bring down 84: 284. Double 7=14, find x: 14x×x≤284. x=2: 142×2=284. Remainder 0. √5184=72 m. (3 marks: 2 for correct long division steps, 1 for answer)
  • A21. 12²=144, 13²=169. Since 144 < 150 < 169, √150 lies between 12 and 13. (2 marks: 1 for identifying squares, 1 for conclusion)

Answer Key: Section E (Long Answer and HOTS Questions)

Detailed solutions for the three long-answer HOTS questions are provided here. These questions demand multi-step reasoning and integration of concepts. For Q22, you must find the largest perfect square ≤2000, then calculate the shortfall. For Q23, algebraic proof demonstrates understanding beyond mere calculation; always state 'Let n be a natural number' to begin proofs. For Q24, verification of Pythagorean triplet requires checking a²+b²=c², and area calculation uses the formula ½×base×height for a right triangle. CBSE marking schemes for such questions allocate one mark for correct problem setup, one or two marks for intermediate steps, and one mark for the final answer. If you missed marks, review each step: did you state what you were proving? Did you show algebraic manipulation clearly? Did you box or underline final answers? Long-answer practice is essential for scoring well in annual CBSE exams, where 4-5 mark questions contribute significantly to the total.
  • A22. √2000 ≈ 44.72. Largest perfect square ≤2000 is 44²=1936. Trees needed: 45²−2000 = 2025−2000 = 25 more trees. (4 marks: 1 for finding n, 1 for 44², 1 for 45², 1 for final answer)
  • A23. Proof: Let consecutive numbers be n and n+1. (n+1)²−n² = n²+2n+1−n² = 2n+1 = n+(n+1). Hence the difference equals the sum. Application: 101²−100² = 101+100 = 201. (5 marks: 2 for algebraic proof, 1 for simplification, 1 for application, 1 for final answer)
  • A24. Verification: 8²+15² = 64+225 = 289 = 17². Yes, it is a Pythagorean triplet. Perimeter = 8+15+17 = 40 units. Area = ½×8×15 = 60 square units. (4 marks: 1 for verification, 1 for perimeter, 1 for area formula, 1 for final area value)

How CBSETUTOR.ai Helps Master Squares and Square Roots

Chapter 5 demands both conceptual clarity and computational accuracy. Students often struggle with long division method or make errors in prime factorisation pairings. CBSETUTOR.ai offers a 24×7 AI tutor that allows Class 8 students to upload photos of worksheet problems and receive instant step-by-step solutions tailored to NCERT methods. Whether your child is stuck on identifying Pythagorean triplets or applying the property of square numbers, the AI breaks down each concept into bite-sized explanations. Priced at a flat ₹999 per month for all classes 6-12, it eliminates the need for expensive home tutors or coaching centres. A 3-day free trial lets parents and students experience personalised doubt resolution before committing. During exam revision, students can practice unlimited questions, get immediate feedback, and track progress topic-wise, ensuring no gap remains in Squares and Square Roots or any other NCERT chapter.
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Frequently asked questions

What is the difficulty level and suggested time for this worksheet?+
This worksheet is of medium-to-hard difficulty, aligned with CBSE Class 8 annual exam standards. The suggested time is 90 minutes for all sections, including the case study. Students should attempt it in one sitting for realistic exam practice.
How many marks is this worksheet out of, and how should I divide my time?+
The worksheet totals approximately 35-40 marks. Allocate 10 minutes for MCQs, 8 minutes for fill-ups, 8 minutes for true/false, 25 minutes for short answers, 25 minutes for long answers, and 10 minutes for the case study. Reserve 4 minutes for final review.
Which NCERT exercises does this worksheet align with?+
This worksheet draws questions from NCERT Class 8 Mathematics Chapter 5 Exercises 5.1, 5.2, 5.3, and 5.4. It covers properties of squares (Ex 5.1), patterns in square numbers (Ex 5.2), square roots by factorisation (Ex 5.3), and long division method (Ex 5.4).
My child struggles with the long division method for square roots. How can we practice more?+
Focus on NCERT Exercise 5.4 first. Practice with 4-digit and 6-digit perfect squares like 2916, 5476, 103041. Use graph paper to maintain column alignment. CBSETUTOR.ai provides step-by-step photo-based guidance for long division, showing each pairing and subtraction clearly during the 3-day free trial.
Are Pythagorean triplets important for CBSE exams?+
Yes. Pythagorean triplets appear regularly in 2-3 mark questions and in integrated problems involving area and perimeter of right triangles. Memorise the formula (2m, m²−1, m²+1) and common triplets like (3,4,5), (5,12,13), (8,15,17), (7,24,25) for quick recall.
What common mistakes do students make in square root problems?+
Common errors include: incorrect pairing of prime factors, forgetting to check if all factors are paired, arithmetic mistakes in long division subtraction steps, and not verifying the final answer by squaring it. Always write factors in exponential form (e.g., 2⁴×3²) to simplify pairing.
How do I verify if my square root answer is correct?+
Square your answer and check if it matches the original number. For example, if you found √3969 = 63, verify by calculating 63² = 3969. For non-perfect squares estimated by long division, the square of your quotient should be very close to the original number.
Can this worksheet be used for CBSE periodic tests preparation?+
Absolutely. The worksheet mirrors the question pattern, difficulty level, and marking scheme of CBSE Periodic Test 2 or Term-2 exams. Teachers can print and distribute it as a class assignment, and parents can use it for weekend revision at home.
What is the importance of properties like 'number of zeros is always even' in perfect squares?+
Such properties help quickly identify or eliminate options in MCQs and reduce calculation time. For instance, 1000 has three zeros (odd), so it cannot be a perfect square without checking further. These shortcuts are valuable under exam time pressure.
Where can I find more worksheets and practice material for Class 8 Mathematics?+
NCERT Exemplar Problems and CBSE sample papers are official sources. Additionally, CBSETUTOR.ai generates unlimited topic-wise practice questions, provides instant solutions, and offers a 3-day free trial at ₹999/month flat for all classes, making it a cost-effective alternative to printed workbooks and coaching notes.

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