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By Karan Singh Bisht
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Updated on 8 May 2026, 14:56 IST
Squares and cubes are important topics in Class 8 Maths. They help students understand powers, exponents, factors, number patterns, algebra, mensuration, and higher-level calculations. A square is formed when a number is multiplied by itself, while a cube is formed when a number is multiplied by itself three times. This worksheet includes concept explanation, practice questions, word problems, and answers to help students revise the topic clearly.
The square of a number is the result obtained when the number is multiplied by itself.
Formula: Square of a number = Number × Number
For example:
| Number | Square |
| 4 | 4 × 4 = 16 |
| 7 | 7 × 7 = 49 |
| 12 | 12 × 12 = 144 |
So, 4² = 16, 7² = 49, and 12² = 144.
A number that is the square of another whole number is called a perfect square.
Examples of perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
The cube of a number is the result obtained when the number is multiplied by itself three times.
Formula: Cube of a number = Number × Number × Number
For example:
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| Number | Cube |
| 2 | 2 × 2 × 2 = 8 |
| 5 | 5 × 5 × 5 = 125 |
| 10 | 10 × 10 × 10 = 1000 |
So, 2³ = 8, 5³ = 125, and 10³ = 1000.
A number that is the cube of another whole number is called a perfect cube.
Examples of perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
The number 1729 — smallest number expressible as sum of two cubes in two different ways.
1729 = 1³ + 12³ = 9³ + 10³
Also Read: Class 8 Squares and Square Roots Worksheet
Choose the correct option (1 mark each)
1. The square of an odd number can be:
(a) 256
(b) 361
(c) 144
(d) 400
Answer: (b) 361 because 19² = 361 and 19 is odd
2. Which of the following will have 1 at its unit place?
(a) (124)²
(b) (79)²
(c) (83)²
(d) (160)²
Answer: (b) (79)² numbers ending in 9 have squares ending in 1
3. Which of the following is not a perfect square?
(a) 361
(b) 1156
(c) 1128
(d) 1681
Answer: (c) 1128 ends in 8, so it cannot be a perfect square
4. A perfect square can never have the following digit at ones place:
(a) 1
(b) 6
(c) 5
(d) 3
Answer: (d) 3 digits 2, 3, 7, 8 never appear at units place of a perfect square
5. The smallest perfect square divisible by 4, 9, and 10 is:
(a) 180
(b) 900
(c) 100
(d) 400
Answer: (b) 900
6. The value of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 is:
(a) 78
(b) 64
(c) 85
(d) 81
Answer: (d) 81
sum of first 9 odd numbers = 9² = 81
7. The value of √(176 + √2401) is:
(a) 15
(b) 14
(c) 17
(d) 16
Answer: (a) 15
Solution: √2401 = 49
√(176 + 49) = √225 = 15
8. The smallest square number divisible by each of 6, 9, and 15 is:
(a) 800
(b) 810
(c) 600
(d) 900
Answer: (b) 810
9. The square root of 0.0576 is:
(a) 2.4
(b) 0.024
(c) 0.0024
(d) 0.24
Answer: (d) 0.24
10. The smallest number by which 1944 must be divided to get a perfect square is:
(a) 2
(b) 6
(c) 3
(d) 9
Answer: (b) 6
11. Greatest perfect cube of a 3-digit number is:
(a) 999
(b) 729
(c) 900
(d) 829
Answer: (b) 729
because 9³ = 729
12. Unit digit of the cube of 7 is:
(a) 3
(b) 7
(c) 9
(d) 1
Answer: (a) 3
because 7³ = 343
13. Which of the following is a perfect cube?
(a) 349
(b) 829
(c) 1729
(d) 1331
Answer: (d) 1331 because 11³ = 1331
14. Cube root of 15625 is:
(a) 35
(b) 25
(c) 15
(d) 26
Answer: (b) 25 because 25³ = 15625
15. The cube of a number can end with:
(a) 0,1,4,5,6,9 only
(b) 0,1,8,9
(c) Any digit 0–9
(d) Only even digits
Answer: (c) Any digit from 0 to 9
16. Which smallest number must be subtracted from 243 to make it a perfect cube?
(a) 9
(b) 18
(c) 19
(d) 27
Answer: (d) 27 because 243 − 27 = 216 = 6³
17. Which of the following is a Hardy-Ramanujan number?
(a) 1729
(b) 2406
(c) 13833
(d) None of these
Answer: (a) 1729
18. The largest 5-digit number which is a perfect square is:
(a) 99999
(b) 99764
(c) 99976
(d) 99856
Answer: (d) 99856 because 316² = 99856
19. By what least number must 675 be multiplied to obtain a perfect cube?
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (a) 5 because 675 × 5 = 3375 = 15³
20. The unit digit of cube root of 571787 is:
(a) 1
(b) 4
(c) 3
(d) 5
Answer: (c) 3 numbers ending in 7 have cube roots ending in 3
(a) Both true, R explains A | (b) Both true, R doesn't explain A | (c) A true, R false | (d) A false, R true
Q1. A:A perfect square number between 30 and 40 is 36.
R:A perfect square is a number that can be expressed as the product of an integer by itself.
Answer: (a) Both A and R are true, and R correctly explains A.
Q2. A:The number of zeros in the square of 200 is 4.
R:The number of zeroes in the square of 50 is 3.
Answer: (c) A is true but R is false — 50² = 2500, which has only 2 zeros, not 3.
Q3. A:1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 121.
R:The sum of first n odd natural numbers is n².
Answer: (d) A is false but R is true — sum of first 10 odd numbers = 10² = 100, not 121.
Q4. A:A number ending with digit 8 can never be a perfect square.
R:Squares of numbers ending in 4 or 6 end in 6.
Answer: (b) Both are true but R is not the correct explanation of A.
Q5. A:125 is a perfect cube.
R:A perfect cube is the result of multiplying the same integer three times.
Answer: (a) Both true, R correctly explains A — 5 × 5 × 5 = 125.
Q6. A:The unit digit of cube of 23 is 9.
R:If a number has 3 in its ones place, its cube has 7 in its ones place.
Answer: (d) A is false but R is true — 3³ = 27, unit digit is 7. So cube of 23 ends in 7, not 9.
Q7. A:1729 is called the Hardy-Ramanujan number.
R:It is the smallest number expressible as the sum of two cubes in two different ways.
Answer: (a) Both true, R correctly explains A. 1729 = 1³ + 12³ = 9³ + 10³.
Q8. A:The cube of any even number is even and the cube of any odd number is odd.
R:An even number always has 2 as a factor; an odd number has no factor of 2, so its cube also has no factor of 2.
Answer: (a) Both true, and R is the correct explanation of A.
Complete the statement with the correct answer
Q1. The pair of partner factors of 3 is ___.
Ans. 1 and 3
Q2. All square numbers end with 0, 1, 4, 5, 6, or 9 at the ___ place.
Ans. Units
Q3. If a number has n zeros at the end, its square will have ___ zeros.
Ans. 2n
Q4. Total natural numbers between 19² and 20² is ___.
Ans. 38
Q5. √28900 = ___
Ans. 170
Q6. The smallest number expressible as sum of two cubes in two ways is ___.
Ans. 1729
Q7. Cube of an even number is also an ___ number.
Ans. Even
Q8. The cube root of a 3-digit number is always a ___ digit number.
Ans. One
Q9. Unit digit of cube root of 6859 is ___.
Ans. 9
Q10. ∛12167 = ___
Ans. 23
Q11. The inverse operation of squaring is ___.
Ans. Square Root
Q12. The smallest multiple of 720 that is a perfect square is ___.
Ans. 3600
State whether each statement is correct or not
| S.no | Statement | Answer |
| 1 | The total number of natural numbers between 5² and 6² is 10. | TRUE |
| 2 | If a number ends in 4, then its square root also ends in 4. | FALSE |
| 3 | The total number of natural numbers between 10² and 11² is 20. | TRUE |
| 4 | The sum of first n odd natural numbers is (n² + 1). | FALSE |
| 5 | If 55² = 3025 then 56² = 3025 + 111. | TRUE |
| 6 | √0.09 = 0.3 | TRUE |
| 7 | 4104 and 13832 are Hardy-Ramanujan numbers. | TRUE |
| 8 | The cube root of an odd number is always an odd number. | TRUE |
| 9 | Cube root of 10648 is 22. | TRUE |
| 10 | A number ending with 4 has its cube root ending in 8. | FALSE |
| 11 | The cube root of a 4-digit number always has 2 digits. | TRUE |
| 12 | The cube of any negative number is positive. | FALSE |
1–2 line answers (2 marks each)
1. If a number contains 3 zeros at the end, how many zeros will its square have?
Answer: The square will have 2 × 3 = 6 zeros at the end.
2. What will be the unit digit of: (a) 17² (b) 28² (c) 45²
Answer: (a) 17² → unit digit of 7² = 49 → 9
(b) 28² - unit digit of 8² = 64 - 4
(c) 45² → unit digit of 5² = 25 - 5
3. Find the number of non-perfect square numbers between 7² and 8².
Answer: Non-perfect squares between n² and (n+1)² = 2n = 2 × 7 = 14
4. What is the value of 79² − 78²?
Answer: Using (a² − b²) = (a + b)(a − b): (79 + 78)(79 − 78) = 157 × 1 = 157
5. Find the value of √(248 + √(51 + √169))
Answer: Step 1: √169 = 13
Step 2: √(51 + 13) = √64 = 8
Step 3: √(248 + 8) = √256 = 16
6. What is the sum of first 30 odd natural numbers?
Answer: Sum = n² = 30² = 900
7. Is 216 a perfect cube? What is its cube root?
Answer: Yes, 216 = 6 × 6 × 6 = 6³. Cube root = 6
8. What is the smallest number by which 1600 must be multiplied to get a perfect cube?
Answer: 5 because 1600 × 5 = 8000 = 20³
1. Express 5³ and 6³ as the sum of consecutive odd numbers.
Answer: 5³ = 125 = 21 + 23 + 25 + 27 + 29
6³ = 216 = 31 + 33 + 35 + 37 + 39 + 41
2. Find the square root of 121 by repeated subtraction method.
Answer: 121−1=120
120−3= 117
117−5= 112
112−7= 105
105−9= 96
96−11= 85
85−13= 72
72−15= 57
57−17= 40
40−19= 21
21−21= 0
We subtracted 11 times ∴ √121 = 11
3. Find the cube root of (−3375)/(−2744).
Answer: ∛(−3375/−2744) = ∛3375 / ∛2744 = 15/14
15³ = 3375 and 14³ = 2744
Answer = 15/14
4. Using prime factorization, identify which are perfect cubes: 128, 343, 1728, 59319.
Answer: 128 = 2⁷ exponent 7 not a multiple of 3 - Not a perfect cube
343 = 7³ - Perfect cube
1728 = 12³ - Perfect cube
59319 = 39³ - Perfect cube
5. Find the smallest number by which 26244 must be divided so that the quotient is a perfect cube. Find the cube root.
Answer: 26244 = 2² × 3⁸. For a perfect cube, divide by 2² × 3² = 36.
26244 ÷ 36 = 729 = 9³ → Divide by 36, cube root = 9
6. Find the number that should be multiplied to 392 to make it a perfect square. Also find the square root.
Answer: 392 = 2³ × 7². Multiply by 2 to pair the extra 2.
392 × 2 = 784 = 28²
Multiply by 2, square root = 28
7. Which of these are cubes of even natural numbers? Find the numbers: 216, 512, 729, 1000.
Answer: 216 = 6³ - Even
512 = 8³ - Even
729 = 9³ - Odd (not even)
1000 = 10³ - Even
5 marks each — detailed steps required
1. 2025 plants are to be planted such that each row has as many plants as the number of rows. Find the number of rows.
Answer: Let number of rows = n → n × n = 2025
Prime factorization: 2025 = 3⁴ × 5² = (3² × 5)² = 45²
√2025 = 45 rows, with 45 plants in each row.
2. Students of Class VIII donated ₹2401 for PM's National Relief Fund. Each student donated as many rupees as the number of students. Find the number of students.
Answer: n × n = 2401 → n = √2401
2401 = 7⁴ = (7²)² → √2401 = 49
Number of students = 49
3. The area of a square field is 101(1/400) sq. m. Find the length of one side.
Answer: Area = 101¹⁄₄₀₀ = 40401/400
Side = √(40401/400) = √40401 / √400 = 201/20 = 10.05 m
4. A general arranged 335250 men in square formation. 9 men were left over. How many were in each row?
Answer: Men forming the square = 335250 − 9 = 335241
√335241 = 579 men in each row.
5. Is 53240 a perfect cube? If not, find the smallest number to divide it to get a perfect cube. Find the cube root.
Answer: 53240 = 2³ × 5 × 11³
The factor 5 does not form a complete triplet.
Divide by 5
53240 ÷ 5 = 10648
= 22³
Cube root = 22
6. Each student contributed money equal to the square of total students. Total collected = ₹421875. Find number of students.
Answer: Let students = n → n × n² = n³ = 421875
∛421875 = ∛(75³) = 75 students
Read each passage carefully and answer the questions — 4–5 marks each
Priya wants to design a square box of area 2916 m². She purchased a big cardboard and has to cut it for making the square box.
Q(i) What will be each side of the square box?
Ans. √2916 = 54 m
Q(ii) Calculate the perimeter of the cardboard to apply ribbon all across.
Ans. 4 × 54 = 216 m
Q(iii) If the cost of ribbon is ₹5 per metre, find the total amount spent.
Ans. 216 × 5 = ₹1080
During dance practice, 6570 students are arranged in rows such that the number of students in each row equals the number of columns. The instructor finds 9 children are left out.
Q(i) What is the total number of students forming a square?
Ans. 6570 − 9 = 6561 = 81² → (c) 6561
Q(ii) How many rows of students in the arrangement?
Ans. √6561 = (a) 81
Q(iii) When arranged for dance, number of students equals:
Ans. (c) Number of rows × Number of columns
Gunjan noticed the number of steps she walked on her smart watch in the evening and found it to be 7744.
Q(i) Is 7744 a perfect cube?
Ans. No — 7744 = 2⁵ × 11², exponents not all multiples of 3
Q(ii) What is the smallest number to multiply to make it a perfect cube?
Ans. 11 — to complete the triplet of 11s
Q(iii) What is the cube root of the resulting number?
Ans. 7744 × 11 = 85184 = 44³ → Cube root = 44
Q(iv) Find the unit digit of cube of 7744.
Ans. Unit digit of 4³ = 64 → Unit digit = 4
A school builds a square playground with area 1764 m². A 2 m wide walking path is constructed all around the playground.
Q(i) What is the length of the side of the playground?
Ans. √1764 = 42 m
Q(ii) After constructing the path, what will be the outer side length?
Ans. 42 + 2 + 2 = 46 m
Q(iii) Find the total area including the path.
Ans. 46² = 2116 m²
Q(iv) Find the area of the walking path.
Ans. 2116 − 1764 = 352 m²
Q(v) Is the area of the path a perfect square?
Ans. No 352 ends in 2, so it cannot be a perfect square
Important formulas for Chapter 1 A Square and A Cube
Sum of First n Odd Numbers
1 + 3 + 5 + ... + (2n−1) = n²
e.g., Sum of first 7 odd numbers = 7² = 49
Non-perfect Squares Between Consecutive Squares
Between n² and (n+1)² = 2n numbers
e.g., Between 5² and 6²: 2×5 = 10 numbers
Difference of Consecutive Squares
(n+1)² − n² = 2n + 1
e.g., 6² − 5² = 2(5) + 1 = 11
Zeros in Square of a Number
n zeros → square has 2n zeros
e.g., 200 has 2 zeros
200² = 40000 has 4 zeros
Hardy-Ramanujan Number
1729 = 1³ + 12³ = 9³ + 10³
Smallest number expressible as sum of 2 cubes in 2 different ways
Square Root by Prime Factorization
Pair all prime factors → product of one from each pair
e.g., √900 = √(2²×3²×5²) = 2×3×5 = 30
| Square | Cube |
| A number multiplied by itself two times | A number multiplied by itself three times |
| Written as n² | Written as n³ |
| Example: 6² = 36 | Example: 6³ = 216 |
| Used to calculate area of a square | Used to calculate volume of a cube |
| Perfect square example: 49 | Perfect cube example: 125 |
| Number | Square | Number | Square |
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
| Number | Cube | Number | Cube |
| 1 | 1 | 9 | 729 |
| 2 | 8 | 10 | 1000 |
| 3 | 27 | 11 | 1331 |
| 4 | 64 | 12 | 1728 |
| 5 | 125 | 13 | 2197 |
| 6 | 216 | 14 | 2744 |
| 7 | 343 | 15 | 3375 |
| 8 | 512 |
Solving this worksheet helps students:
| S.No | Benefits |
| 1 | Understand the difference between squares and cubes |
| 2 | Improve calculation speed |
| 3 | Identify perfect squares and perfect cubes easily |
| 4 | Strengthen concepts of exponents and powers |
| 5 | Prepare better for school exams |
| 6 | Build a strong base for algebra and number systems |
| 7 | Practice application-based word problems |
Students can use these tips to solve squares and cubes questions quickly:
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A Square and A Cube Class 8 Maths Chapter 1 Worksheet covers important topics such as perfect squares, perfect cubes, square roots, cube roots, properties of squares and cubes, unit digit patterns, and word problems. Students can also use Infinity Learn worksheets to practise chapter-wise questions with answers.
Students should solve this worksheet to improve calculation speed, understand number patterns, and strengthen their concepts of squares and cubes. Regular practice helps students prepare better for school exams. Infinity Learn provides practice-based resources that help students revise the chapter more effectively.
A perfect square is a number obtained by multiplying a number by itself, such as 36 = 6². A perfect cube is a number obtained by multiplying a number three times, such as 125 = 5³. Students can also use prime factorisation to check whether a number is a perfect square or a perfect cube. Infinity Learn’s step-by-step solutions can help students understand this method clearly.
Yes, the worksheet includes answers with explanations so students can check their solutions and correct mistakes. Step-by-step answers help improve understanding and revision. Students can also refer to Infinity Learn for Class 8 Maths worksheets, NCERT solutions, and chapter-wise practice questions.