A Square and A Cube — Class 8 Mathematics (Ganita Prakash)
"1, 4, 9, 16, 25 — what do you see growing from one square to the next? You are looking at the first language humans used to count."
1. About the Chapter
'A Square and A Cube' is the opening chapter of the new NCERT Class 8 Mathematics textbook Ganita Prakash (2025-26 onwards). True to the textbook's NEP 2020 spirit, the chapter does NOT just dump formulas — it lets students discover the pattern of squares and cubes through geometry, dots, and history.
Quick Snapshot
- What you learn: Perfect squares, square roots, perfect cubes, cube roots
- Major skills: Identifying squares/cubes; finding roots via prime factorisation; estimating roots
- Geometric insight: Why a 'square' looks like a square, and a 'cube' looks like a cube
- Indian historical context: From the Sulba Sutras to Aryabhata to modern mathematics
Key Idea
A square number is what you get when you multiply a number by itself once (n × n = n²). A cube number is what you get when you multiply a number by itself twice (n × n × n = n³).
The names 'square' and 'cube' are not random — they describe the shapes these numbers physically make.
2. Squares — The Foundation
Definition
A square number (or perfect square) is the product of a whole number with itself.
If n is any integer, then n²= n × n is a square number.
Examples
| n | n² | Visual (dots) |
|---|---|---|
| 1 | 1 | • |
| 2 | 4 | 2×2 grid (4 dots) |
| 3 | 9 | 3×3 grid (9 dots) |
| 4 | 16 | 4×4 grid (16 dots) |
| 5 | 25 | 5×5 grid (25 dots) |
Why 'Square'?
If you arrange dots in a perfect square shape — equal rows and columns — the total dots will always be a square number. The geometry IS the algebra.
First Twenty Square Numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
Properties of Square Numbers
- A square number always ends in 0, 1, 4, 5, 6, or 9 (never in 2, 3, 7, or 8).
- A square number cannot end in an odd number of zeros (e.g., 100 = 10² is OK, but 1000 cannot be a perfect square).
- Square of an even number is even; square of an odd number is odd.
- The square of a positive number is positive; the square of a negative number is also positive (e.g., (−3)² = 9).
- Sum of first n odd numbers = n².
- 1 = 1²
- 1 + 3 = 4 = 2²
- 1 + 3 + 5 = 9 = 3²
- 1 + 3 + 5 + 7 = 16 = 4²
- This is one of the most beautiful patterns in mathematics.
Patterns in Square Numbers
Pattern 1: Differences Differences between consecutive squares form odd numbers:
- 4 − 1 = 3
- 9 − 4 = 5
- 16 − 9 = 7
- 25 − 16 = 9
- The differences are 3, 5, 7, 9 — the odd numbers in order.
Pattern 2: Squares of numbers ending in 5 Any number ending in 5: n5² = n(n+1) × 100 + 25.
- 25² = 2(3)|25 = 625
- 35² = 3(4)|25 = 1225
- 45² = 4(5)|25 = 2025
- 95² = 9(10)|25 = 9025
Pattern 3: Squaring with the Indian Vedic method (Multiplication shortcuts) For two-digit numbers, several Vedic tricks apply — explored in the optional 'History of Maths' boxes in Ganita Prakash.
3. Square Roots
Definition
A square root of a number is the number which, when squared, gives the original.
If n² = m, then n is the square root of m, written as √m.
So √25 = 5 because 5 × 5 = 25.
Methods to Find Square Roots
Method 1: Prime Factorisation Method
Step 1: Find the prime factorisation of the number. Step 2: Pair the prime factors. Step 3: For each pair, take one factor. Step 4: Multiply the chosen factors together.
Example: Find √324.
- 324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 2) × (3 × 3) × (3 × 3)
- Take one from each pair: 2 × 3 × 3 = 18
- ∴ √324 = 18 ✓ (Check: 18² = 324)
Example: Find √576.
- 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)
- Take one from each pair: 2 × 2 × 2 × 3 = 24
- ∴ √576 = 24 ✓
Method 2: Division Method (Long Division)
For larger numbers where prime factorisation is hard, use the long-division method.
Example: Find √1849.
- Step 1: Pair digits from right: 18|49.
- Step 2: Largest square ≤ 18 is 16 (= 4²). Write 4 above; remainder 18 − 16 = 2.
- Step 3: Bring down next pair (49) → 249.
- Step 4: Double quotient: 4 × 2 = 8. Find digit x such that 8x × x ≤ 249.
- Try x = 3: 83 × 3 = 249 ✓
- Step 5: Quotient is 43. So √1849 = 43.
When a Number is NOT a Perfect Square
Numbers like 10, 12, 27, 50 are not perfect squares. Their square roots are irrational numbers (non-terminating, non-repeating decimals):
- √2 ≈ 1.41421
- √3 ≈ 1.73205
- √5 ≈ 2.23607
These were studied by the ancient Indian mathematician Baudhayana in the Sulba Sutras (~800 BCE).
4. Cubes — Stepping Up to Three Dimensions
Definition
A cube number (or perfect cube) is the product of a whole number used three times.
If n is any integer, then n³ = n × n × n is a cube number.
Examples
| n | n³ | Why 'cube' |
|---|---|---|
| 1 | 1 | 1×1×1 single dot/unit |
| 2 | 8 | 2×2×2 stacked cube (8 unit cubes) |
| 3 | 27 | 3×3×3 stacked cube (27 unit cubes) |
| 4 | 64 | 4×4×4 stacked cube |
| 5 | 125 | 5×5×5 stacked cube |
Why 'Cube'?
A cube number is the number of unit cubes needed to build a perfect 3D cube of side n.
First Twenty Cube Numbers
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000.
Properties of Cubes
- Cube of an even number is even; cube of an odd number is odd.
- Cube of a positive number is positive; cube of a negative number is negative: (−3)³ = −27.
- Cube of a number ending in 0 ends in three zeros: 10³ = 1000.
- Cubes can end in any digit (0–9) — unlike squares.
- Sum of consecutive odd numbers gives consecutive cubes:
- 1 = 1 = 1³
- 3 + 5 = 8 = 2³
- 7 + 9 + 11 = 27 = 3³
- 13 + 15 + 17 + 19 = 64 = 4³
Pattern — Cubes and Triangular Numbers
Sum of cubes of first n natural numbers = (sum of first n natural numbers)².
- 1³ + 2³ = 9 = (1+2)² = 3² ✓
- 1³ + 2³ + 3³ = 36 = (1+2+3)² = 6² ✓
- 1³ + 2³ + 3³ + 4³ = 100 = 10² ✓
This is Nicomachus's theorem — beautiful and ancient.
5. Cube Roots
Definition
A cube root of a number is the number which, when cubed, gives the original.
If n³ = m, then n is the cube root of m, written as ∛m.
So ∛125 = 5 because 5 × 5 × 5 = 125.
Method: Prime Factorisation for Cube Roots
Step 1: Find prime factorisation of the number. Step 2: Group prime factors in triplets (groups of 3). Step 3: From each triplet, take one factor. Step 4: Multiply chosen factors.
Example: Find ∛216.
- 216 = 2 × 2 × 2 × 3 × 3 × 3 = (2³) × (3³)
- Take one from each triplet: 2 × 3 = 6
- ∴ ∛216 = 6 ✓ (Check: 6³ = 216)
Example: Find ∛1728.
- 1728 = 2³ × 2³ × 3³ = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
- = (2³)(2³)(3³)
- Take one from each triplet: 2 × 2 × 3 = 12
- ∴ ∛1728 = 12 ✓
Example: Find ∛729.
- 729 = 3³ × 3³ = 3 × 3 × 3 × 3 × 3 × 3
- Take one from each triplet: 3 × 3 = 9
- ∴ ∛729 = 9 ✓
Estimation of Cube Roots
For 3-digit cubes, the cube root has 1 digit; for cubes between 1000 and 10⁶, the cube root has 2 digits. Use this to estimate.
6. Historical Context — Indian Mathematics
Sulba Sutras (~800 BCE)
- Earliest Indian mathematical texts
- Used by priests to construct fire altars (vedis)
- Required precise calculations of square roots
- Approximated √2 to high accuracy: √2 ≈ 1.41421356 (correct to 7 decimal places!)
Aryabhata (476–550 CE)
- Wrote 'Aryabhatiya' (~499 CE)
- Gave methods for finding square roots and cube roots
- One of the earliest known algorithms for cube roots
Brahmagupta (598–668 CE)
- Worked on quadratic equations, square roots, and cube roots
- His work was translated into Arabic and reached medieval Europe
Indian Influence
Indian methods for squares, cubes, square roots, and cube roots travelled west via Arabic mathematics — and eventually became part of the global mathematical tradition.
7. Key Formulas and Identities
Square Identities
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
Cube Identities
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a − b)³ = a³ − 3a²b + 3ab² − b³ — Wait, let me restate properly:
- (a − b)³ = a³ − 3a²b + 3ab² − b³
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
Quick Reference Squares (Up to 30²)
1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, 18²=324, 19²=361, 20²=400, 21²=441, 22²=484, 23²=529, 24²=576, 25²=625, 26²=676, 27²=729, 28²=784, 29²=841, 30²=900.
Quick Reference Cubes (Up to 15³)
1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000, 11³=1331, 12³=1728, 13³=2197, 14³=2744, 15³=3375.
8. Worked Examples
Example 1: Identify a Perfect Square
Is 1024 a perfect square?
- 1024 = 2¹⁰
- Even number of 2s → pairable
- 1024 = (2⁵)² = 32² ✓
- Yes, √1024 = 32.
Example 2: Identify a Perfect Cube
Is 392 a perfect cube?
- 392 = 2³ × 7² = 2³ × 49
- Has a triplet of 2s but only TWO 7s
- Not a perfect cube.
Example 3: Make a Number a Perfect Square
What is the smallest number by which 252 must be multiplied to get a perfect square?
- 252 = 2² × 3² × 7
- Pairs: (2×2), (3×3); 7 is unpaired
- Multiply by 7 to pair it: 252 × 7 = 1764 = 42² ✓
Example 4: Make a Number a Perfect Cube
What is the smallest number by which 392 must be multiplied to get a perfect cube?
- 392 = 2³ × 7²
- Need one more 7 to make a triplet
- Multiply by 7: 392 × 7 = 2744 = 14³ ✓
Example 5: Estimate √2025
- Between 40² (=1600) and 50² (=2500)
- Closer to 45² (=2025) — exact!
- √2025 = 45
9. Common Mistakes and How to Avoid Them
-
Squaring vs Doubling:
- 5² = 25 (not 10!)
- Squaring means MULTIPLYING by itself, not DOUBLING.
-
Cubing vs Tripling:
- 4³ = 64 (not 12!)
- Cubing means multiplying THREE TIMES.
-
Square root of negative numbers:
- In Class 8, √(−4) is NOT defined in real numbers.
- Negative cube root IS defined: ∛(−27) = −3.
-
Mis-pairing in prime factorisation:
- Always check: square root needs PAIRS; cube root needs TRIPLETS.
-
Forgetting both positive and negative roots:
- When solving x² = 49, x = ±7 (both 7 and −7).
10. Real-World Applications
Architecture and Construction
- Sulba Sutras used square roots for fire-altar geometry
- Modern buildings use square calculations for area
- Cubes used for volume of containers
Computing
- Hash functions use squaring
- Cryptography (RSA) relies on squares and roots of huge numbers
- Cube roots appear in 3D graphics
Physics
- Kinetic energy = ½ mv² (uses square)
- Volume of cubes = side³
Everyday
- Square plots of land — area in square metres
- Cubic capacity of car engines — measured in cc (cubic centimetres)
- Surface area of a cube = 6a²; volume = a³
11. Tips for Mastery
For Exam Preparation
- Memorise squares up to 30² and cubes up to 15³ — saves time in exams.
- Master prime factorisation — the universal tool for both squares and cubes.
- Practise long-division method for non-obvious square roots.
- Recognise patterns — sum of odd numbers, differences, etc.
- Always check the answer by squaring/cubing back.
Practice Volume
- At least 15-20 problems of each type
- Mix easy, medium, hard difficulty
- Include word problems (smallest number to multiply/divide)
12. Conclusion
The chapter 'A Square and A Cube' is much more than a list of formulas. Ganita Prakash teaches you to see squares and cubes — as geometric shapes, as patterns, as historical achievements of Indian mathematics.
By mastering this chapter, you gain:
- Computational tools (find squares, cubes, roots quickly)
- Pattern recognition (sum of odd numbers, Nicomachus's theorem)
- Historical perspective (Sulba Sutras, Aryabhata, Brahmagupta)
- Geometric intuition (what 'squaring' and 'cubing' really mean)
These ideas will be used throughout your Class 8 maths journey — and beyond, into Class 9, 10, and higher math. Master them now, and the future chapters become so much easier.
