Class 8 : Maths – Lesson 1. A Square and A Cube
EXPLANATION AND ANALYSIS
🌍 INTRODUCTION — SHAPES IN OUR SURROUNDINGS
🏠 When we observe objects around us carefully, we notice many regular shapes.
🎲 A dice, 🧊 an ice cube, 🧱 a box, 📒 a notebook, 🧩 floor tiles — all these objects are closely related to squares and cubes.
🧠 Mathematics helps us:
recognise these shapes
understand their properties
measure them correctly
apply them in daily life
📘 This lesson introduces two fundamental shapes:
Square → a flat shape
Cube → a solid shape
These ideas form the base of geometry and mensuration.
📐 THE SQUARE — BASIC CONCEPT
🔷 A square is a plane or two-dimensional figure.
🔷 It lies flat on a surface and has only length and breadth.
✨ Main idea:
A square has four sides
All sides are equal
Each angle is a right angle (90°)
🧩 Because of equal sides and equal angles, a square is a very balanced and symmetric shape.
🔍 PROPERTIES OF A SQUARE
🟢 All four sides are equal in length
🟣 All angles are right angles
🟡 Opposite sides are parallel
🔵 Diagonals are equal
🟥 Diagonals bisect each other at right angles
📌 These properties help us identify and use squares easily in problems.
📏 PERIMETER OF A SQUARE
📍 The perimeter of a square means the total length of its boundary.
🔵 Formula for perimeter
If side = a,
Perimeter = a + a + a + a
Perimeter = 4 × a
🧾 Example
If side = 9 cm,
Perimeter = 4 × 9
Perimeter = 36 cm
📐 Perimeter is always written in units of length such as cm or m.
📐 AREA OF A SQUARE
📍 The area of a square tells us how much surface it covers.
🔵 Formula for area
If side = a,
Area = a × a = a²
🧾 Example
If side = 7 cm,
Area = 7 × 7
Area = 49 cm²
🟩 Area is always written in square units.
🧊 INTRODUCTION TO A CUBE
🧱 A cube is a solid or three-dimensional shape.
📦 It has length, breadth, and height.
🌐 Key idea:
All edges of a cube are equal
All faces are squares
🧊 Common examples:
dice
sugar cube
ice cube
gift box
🔶 PARTS OF A CUBE
🧩 Faces
A cube has 6 faces, and each face is a square.
🔗 Edges
A cube has 12 edges, all of equal length.
📍 Vertices
A cube has 8 vertices, also called corners.
📘 Knowing these parts helps us understand solid shapes clearly.
📐 SURFACE AREA OF A CUBE
📍 The surface area of a cube is the total area of all its faces.
🔵 Formula for surface area
If side = a,
Surface Area = 6 × a²
🧾 Example
If side = 5 cm,
Area of one face = 5² = 25 cm²
Surface Area = 6 × 25
Surface Area = 150 cm²
📐 Surface area is written in square units.
📦 VOLUME OF A CUBE
🔵 Formula for volume
If side = a,
Volume = a × a × a = a³
🧾 Example
If side = 4 cm,
Volume = 4 × 4 × 4
Volume = 64 cm³
📦 Volume is written in cubic units.
⚖️ DIFFERENCE BETWEEN SQUARE AND CUBE
🔷 Square
flat shape
two-dimensional
has perimeter and area
does not occupy space
🧊 Cube
solid shape
three-dimensional
has surface area and volume
occupies space
🧠 This clear difference avoids confusion between formulas.
🏠 USES IN DAILY LIFE
🧩 Uses of squares
tiles and flooring
chessboards
drawing grids
fields and plots
📦 Uses of cubes
packing boxes
storage containers
building blocks
construction models
🌍 Geometry helps us understand and design the world around us.
⚠️ COMMON MISTAKES TO AVOID
🚫 Mixing perimeter and area formulas
🚫 Writing wrong units
🚫 Forgetting number of faces of a cube
🚫 Confusing a² with a³
✔️ Always check:
formula
calculation
unit
🌟 IMPORTANCE OF THIS LESSON
🏆 Builds strong foundation of geometry
🚀 Prepares for mensuration topics
🧠 Improves spatial understanding
📘 Useful for higher classes
🌱 Connected with real-life situations
🧾 SUMMARY
📐 A square has four equal sides and right angles
📏 Perimeter of a square = 4 × side
🟩 Area of a square = side²
🧊 A cube has six square faces
📐 Surface area of a cube = 6 × side²
📦 Volume of a cube = side³
🔁 QUICK RECAP
🧩 Square → flat figure
🧊 Cube → solid figure
📐 Area → surface covered
📦 Volume → space occupied
🧮 Correct formula + correct unit = correct answer
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TEXTBOOK QUESTIONS
🔒 ❓ Q1. Find the cube roots of 27000 and 10648.
📌 ✅ Answer:
🟢 Step 1
⬥ 27000 = 27 × 1000 = 3³ × 10³ = (3×10)³
⬥ Cube root of 27000 = 30
🔵 Step 2
⬥ 10648 = 22 × 22 × 22 = 22³
⬥ Cube root of 10648 = 22
🔒 ❓ Q2. What number will you multiply by 1323 to make it a cube number?
📌 ✅ Answer:
🟢 Step 1
⬥ 1323 = 3 × 3 × 3 × 7 × 7 = 3³ × 7²
🔵 Step 2
⬥ To make a cube, each prime power must be a multiple of 3
⬥ 7² needs one more 7
⬥ Required number = 7
🔒 ❓ Q3. State true or false. Explain your reasoning.
🔒 ❓ (a) The cube of any odd number is even.
📌 ✅ Answer:
⬥ Odd × odd × odd = odd
⬥ Example: 3³ = 27
⬥ False
🔒 ❓ (b) There is no perfect cube that ends with 8.
📌 ✅ Answer:
⬥ 2³ = 8 ends with 8
⬥ Also, 12³ = 1728 ends with 8
⬥ False
🔒 ❓ (c) The cube of a 2-digit number may be a 3-digit number.
📌 ✅ Answer:
⬥ Smallest 2-digit number is 10
⬥ 10³ = 1000, which has 4 digits
⬥ False
🔒 ❓ (d) The cube of a 2-digit number may have seven or more digits.
📌 ✅ Answer:
⬥ Largest 2-digit number is 99
⬥ 99³ = 970299, which has 6 digits
⬥ False
🔒 ❓ (e) Cube numbers have an odd number of factors.
📌 ✅ Answer:
⬥ Only perfect squares have an odd number of factors
⬥ Example: 8 has factors 1, 2, 4, 8 (even number)
⬥ False
🔒 ❓ Q4. Guess cube roots without factorisation: 1331, 4913, 12167, 32768.
📌 ✅ Answer:
🟢 Step 1
⬥ Use the unit-digit pattern of cubes
🔵 Step 2
⬥ 11³ = 1331 ⟹ cube root = 11
⬥ 17³ = 4913 ⟹ cube root = 17
⬥ 23³ = 12167 ⟹ cube root = 23
⬥ 32³ = 32768 ⟹ cube root = 32
🔒 ❓ Q5. Which of the following is the greatest? Explain.
(i) 67³ − 66³ (ii) 43³ − 42³ (iii) 67² − 66² (iv) 43² − 42²
📌 ✅ Answer:
🟢 Step 1
⬥ Use identities:
⬥ a³ − b³ = (a − b)(a² + ab + b²)
⬥ a² − b² = (a − b)(a + b)
🔵 Step 2
⬥ Here, (a − b) = 1 in all cases
⬥ 67³ − 66³ = 67² + 67×66 + 66² = 13267
⬥ 43³ − 42³ = 5419
⬥ 67² − 66² = 133
⬥ 43² − 42² = 85
⬥ Greatest is (i) 67³ − 66³
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OTHER IMPORTANT QUESTIONS
🔹 PART A — MCQs
🔒 ❓ Question 1.
A number is multiplied by 2² × 3 × 5 to become a perfect cube. Which factor is the minimum required to be multiplied further?
🟢1️⃣ 2 × 3² × 5²
🔵2️⃣ 2 × 3 × 5
🟡3️⃣ 2² × 3² × 5
🟣4️⃣ 3² × 5²
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 2.
Which statement is always true for any natural number n?
🟢1️⃣ (n + 1)³ − n³ is always odd
🔵2️⃣ (n + 1)³ − n³ increases as n increases
🟡3️⃣ (n + 1)³ − n³ is constant
🟣4️⃣ (n + 1)³ − n³ equals 3n²
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 3.
A cube number ends with digit 7. What can be said about its cube root?
🟢1️⃣ It must end with 3
🔵2️⃣ It must end with 7
🟡3️⃣ It must be even
🟣4️⃣ It must be divisible by 3
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 4.
Which of the following cannot be a perfect cube?
🟢1️⃣ 2⁶ × 5³
🔵2️⃣ 3³ × 7³
🟡3️⃣ 2⁴ × 3³
🟣4️⃣ 11³
✔️ Answer: 🟡3️⃣
🔒 ❓ Question 5.
If n is a two-digit natural number, which is impossible for n³?
🟢1️⃣ A five-digit number
🔵2️⃣ A six-digit number
🟡3️⃣ A seven-digit number
🟣4️⃣ A three-digit number
✔️ Answer: 🟣4️⃣
🔒 ❓ Question 6.
Which difference is the greatest?
🟢1️⃣ 54³ − 53³
🔵2️⃣ 68² − 67²
🟡3️⃣ 43³ − 42³
🟣4️⃣ 99² − 98²
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 7.
A number has exactly 27 factors. What is the strongest conclusion?
🟢1️⃣ It must be a perfect cube
🔵2️⃣ It must be a perfect square
🟡3️⃣ It must be prime
🟣4️⃣ It must be even
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 8.
Without factorisation, the cube root of 17576 can be identified because:
🟢1️⃣ It lies between 25³ and 27³
🔵2️⃣ It ends with digit 6
🟡3️⃣ It is divisible by 8
🟣4️⃣ It is a square number
✔️ Answer: 🟢1️⃣
🔒 ❓ Question 9.
Which expression grows faster as n increases?
🟢1️⃣ n²
🔵2️⃣ n³
🟡3️⃣ 3n
🟣4️⃣ 2n²
✔️ Answer: 🔵2️⃣
🔒 ❓ Question 10.
If a number is both a perfect square and a perfect cube, it must be:
🟢1️⃣ A sixth power
🔵2️⃣ A fourth power
🟡3️⃣ A third power
🟣4️⃣ A square of a prime
✔️ Answer: 🟢1️⃣
🔹 PART B — Short Answer Questions
🔒 ❓ Question 11.
Why does every perfect cube have an odd number of factors?
📌 ✅ Answer:
🔹 In a perfect cube, all prime factors occur in groups of three
🔹 Pairing of factors leaves exactly one unpaired factor
🔒 ❓ Question 12.
Explain why no cube number ends with digit 8.
📌 ✅ Answer:
🔹 Units digits of cubes follow fixed patterns
🔹 No natural number cubed ends with digit 8
🔒 ❓ Question 13.
Decide whether 46656 is a perfect cube without factorisation.
📌 ✅ Answer:
🔹 36³ = 46656
🔹 Therefore, 46656 is a perfect cube
🔒 ❓ Question 14.
Why is (n + 1)³ − n³ always increasing for natural numbers n?
📌 ✅ Answer:
🔹 (n + 1)³ − n³ = 3n² + 3n + 1
🔹 This expression increases as n increases
🔒 ❓ Question 15.
Compare the growth of square numbers and cube numbers.
📌 ✅ Answer:
🔹 Cube numbers grow faster than square numbers
🔹 Power 3 increases magnitude more rapidly than power 2
🔒 ❓ Question 16.
Explain why a two-digit number cannot have a three-digit cube.
📌 ✅ Answer:
🔹 The smallest two-digit number is 10
🔹 10³ = 1000, which has four digits
🔒 ❓ Question 17.
Why must prime factors appear in triples for a number to be a perfect cube?
📌 ✅ Answer:
🔹 A cube is formed by multiplying three identical numbers
🔹 Hence each prime factor must occur three times
🔒 ❓ Question 18.
Decide whether 250047 is a perfect cube by estimation.
📌 ✅ Answer:
🔹 63³ = 250047
🔹 Hence, 250047 is a perfect cube
🔒 ❓ Question 19.
Explain why 1 is both a perfect square and a perfect cube.
📌 ✅ Answer:
🔹 1 = 1² and also 1 = 1³
🔹 It has exactly one factor
🔒 ❓ Question 20.
Why is factor-count logic useful in identifying cube numbers?
📌 ✅ Answer:
🔹 Perfect cubes always have an odd number of factors
🔹 This helps eliminate non-cube numbers quickly
🔹 PART C — Detailed Answer Questions
🔒 ❓ Question 21.
Show that 91125 is a perfect cube.
📌 ✅ Answer:
🔹 45³ = 91125
🔹 Therefore, 91125 is a perfect cube
🔒 ❓ Question 22.
Without factorisation, find the cube root of 148877.
📌 ✅ Answer:
🔹 53³ = 148877
🔹 Cube root = 53
🔒 ❓ Question 23.
Explain why (n + 1)³ − n³ is always greater than (n + 1)² − n².
📌 ✅ Answer:
🔹 (n + 1)³ − n³ = 3n² + 3n + 1
🔹 (n + 1)² − n² = 2n + 1
🔹 For n ≥ 1, the cubic difference is greater
🔒 ❓ Question 24.
Prove that the cube of any odd number is odd.
📌 ✅ Answer:
🔹 Let n = 2k + 1
🔹 (2k + 1)³ is always odd
🔒 ❓ Question 25.
Find the smallest number by which 1800 must be multiplied to make it a perfect cube.
📌 ✅ Answer:
🔹 1800 = 2³ × 3² × 5²
🔹 Required factor = 3 × 5
🔹 Smallest number = 15
🔒 ❓ Question 26.
Explain why a number having exactly 16 factors cannot be a perfect cube.
📌 ✅ Answer:
🔹 Perfect cubes have an odd number of factors
🔹 16 is even, so the number cannot be a cube
🔒 ❓ Question 27.
Compare 59³ − 58³ and 99² − 98² using reasoning only.
📌 ✅ Answer:
🔹 Differences of cubes grow faster than differences of squares
🔹 Hence, 59³ − 58³ is greater
🔒 ❓ Question 28.
Justify that 262144 is a perfect cube.
📌 ✅ Answer:
🔹 64³ = 262144
🔹 Therefore, 262144 is a perfect cube
🔒 ❓ Question 29.
Explain why cube roots can be estimated using nearest known cube numbers.
📌 ✅ Answer:
🔹 Cube numbers increase steadily
🔹 Nearest cubes give a reliable estimation range
🔒 ❓ Question 30.
A number is both a perfect square and a perfect cube. Explain why it must be a sixth power.
📌 ✅ Answer:
🔹 Square ⇒ power 2, cube ⇒ power 3
🔹 LCM of 2 and 3 is 6
🔹 Hence, the number is of the form a⁶
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