Download A Square and A Cube – NCERT Class 8 Maths Chapter 1 PDF from the new NCERT Class 8 Maths book Ganita Prakash (2026-27). Read the chapter online or download the free PDF. A locker is toggled once for each factor of its number, so it ends up open only when the factor count is odd. Factors come in partner pairs, and a pair repeats itself only in a square — so exactly the square-numbered lockers stay open.
A Square and A Cube PDF – Chapter 1 of Ganita Prakash (Part I)
Read the chapter online or download the PDF.
| Chapter | Download | |
|---|---|---|
| Chapter 1 PDF | A Square and A Cube | View PDF |
| Full book | NCERT Class 8 Maths – Ganita Prakash | View Book |
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What Is in A Square and A Cube? (Class 8 Maths Chapter 1)
- A locker is toggled once for each factor of its number, so it ends up open only when the factor count is odd. Factors come in partner pairs, and a pair repeats itself only in a square — so exactly the square-numbered lockers stay open.
- A perfect square is n × n = n2 for a natural number n. Perfect squares end only in 0, 1, 4, 5, 6 or 9, and can carry only an even number of terminal zeros.
- The sum of the first n odd numbers is n2. Subtracting 1, 3, 5, … in turn is therefore a test: if you land exactly on 0 at the k-th step, the number is k2.
- Prime factorisation settles both questions. A number is a perfect square when its prime factors split into two identical groups, and a perfect cube when they split into three.
- Square root is the inverse of squaring; every perfect square has two integer roots, + and –, and this chapter uses only the positive one, written √. Cube root is written ∛ and a cube has just one real cube root.
- Successive differences of the squares become constant at level 2 and of the cubes at level 3, a first taste of how the degree of a pattern shows itself — and Sanskrit named all of this long ago: varga for square, ghana for cube, mula (root of a plant) for the root operation, the ancestor of the Arabic jidhr and the Latin radix.
Quick revision – A Square and A Cube
| Idea | Meaning | Test / rule | Example |
|---|---|---|---|
| Square number | N × n, written n2 | Factors split into two identical groups | 182 = 324 = (2×3×3)2 |
| Units digit | Last digit of a square | Only 0, 1, 4, 5, 6, 9 possible | 1027 ends in 7 → not a square |
| Terminal zeros | Zeros at the end of a square | Always an even count | 4002 = 160000 (4 zeros) |
| Odd-number sum | 1 + 3 + 5 + … + (2n–1) | Equals n2 | 1+3+5+7+9 = 25 = 52 |
| N-th odd number | 2n – 1 | 36th odd number = 71 | 352 + 71 = 362 = 1296 |
| Gap between squares | Count of numbers between n2 and (n+1)2 | 2n | Between 162 and 172: 32 numbers |
| Triangular link | Tn–1 + Tn | Equals n2 | 10 + 15 = 25 = 52 |
| Square root | X with x2 = y | √y; two integer roots, ± | √441 = 21 |
| Cube number | N × n × n, written n3 | Factors split into three identical groups | 3375 = (3×5)3 = 153 |
| Cube units digit | Last digit of a cube | All ten digits 0–9 occur | 23 = 8, 83 = 512 |
| Cube root | X with x3 = y | ∛y | ∛10648 = 22 |
| Successive differences | Repeated differencing | Constant at level 2 for squares, level 3 for cubes | Cubes: 6, 6, 6, … |
Old NCERT vs New – Class 8 Maths
This chapter combines two chapters of the old NCERT Class 8 Maths book – “Squares and Square Roots” and “Cubes and Cube Roots”.
Solutions and All Chapters
Answers to the exercises and the full chapter list of Ganita Prakash:
A Square and A Cube – Overview
| Class | Class 8 |
| Subject | Maths |
| Book | Ganita Prakash – new NCERT book |
| Part | Part I |
| Chapter | 1 of 14 – A Square and A Cube |
| Book pages | 1 – 18 |
| Edition | NCERT Reprint 2026-27 |
| Session | 2026-27 |
| Free – read online or download |
Class 8 Maths Chapter 1 – FAQs
Where can I download A Square and A Cube Class 8 PDF?
From the download table on this page – the chapter PDF is free, and the full book is linked too.
What is Class 8 Maths Chapter 1 about?
A locker is toggled once for each factor of its number, so it ends up open only when the factor count is odd. Factors come in partner pairs, and a pair repeats itself only in a square — so exactly the square-numbered lockers stay open.
Is A Square and A Cube from the new NCERT Class 8 Maths book?
Yes. It is chapter 1 of ganita prakash (part i), the new NCERT Class 8 Maths book for 2026-27.
Is this the latest NCERT edition for 2026-27?
Yes – the PDF is NCERT’s latest edition for the 2026-27 session.
Where are the NCERT Solutions for Class 8 Maths Chapter 1?
Open the NCERT Solutions link above for answers to the exercise questions.