Chapter 8 of 14
Chapter 8: Carts and Wheels
Circles, Centre, Radius, Diameter, Compass Drawing & Spinning Tops (Charkhi) | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- Identifying round circular objects: bullock cart wheels, bicycle wheels, bangles, coins, chapati.
- Key circle vocabulary: Centre, Radius (r), Diameter (d), and Circumference.
- Relationship between radius and diameter: Diameter = 2 × Radius; Radius = Diameter ÷ 2.
- Technique for drawing a circle using a nail and string, or a geometric compass.
- Making spinning tops (charkhi) and balancing the stick at the exact centre.
All Textbook Exercises & In-Text Tasks (15 Questions)
100% Complete Ingestion
Q1
Radius and Diameter Relationships
Fill in the blanks: (a) If the radius of a circle is 4 cm, its diameter is __ cm. (b) If the diameter of a wheel is 18 cm, its radius is __ cm. (c) Which circle is larger: Circle A with radius 6 cm or Circle B with diameter 10 cm?
[3 Marks]
✅ Step-by-Step Solution & Working
(a) Formula: Diameter = 2 × Radius
• Diameter = 2 × 4 cm = 8 cm.
(b) Formula: Radius = Diameter ÷ 2
• Radius = 18 cm ÷ 2 = 9 cm.
(c) Comparison between Circle A and Circle B:
• Circle A radius = 6 cm (so Diameter = 2 × 6 = 12 cm).
• Circle B diameter = 10 cm (so Radius = 10 ÷ 2 = 5 cm).
• Since 6 cm > 5 cm, Circle A is larger.
• Diameter = 2 × 4 cm = 8 cm.
(b) Formula: Radius = Diameter ÷ 2
• Radius = 18 cm ÷ 2 = 9 cm.
(c) Comparison between Circle A and Circle B:
• Circle A radius = 6 cm (so Diameter = 2 × 6 = 12 cm).
• Circle B diameter = 10 cm (so Radius = 10 ÷ 2 = 5 cm).
• Since 6 cm > 5 cm, Circle A is larger.
CBSE Step-by-Step Marking Scheme
Correct answers for (a) and (b)
1.5 M
Accurate comparison showing Circle A is larger
1.5 M
Q2
Spinning Top Physics & Circle Centre
Zakir, Appu, Naina, and Guddo made cardboard spinning tops. Zakir put his stick near the edge, Appu put it halfway between center and edge, Naina put it slightly off-center, while Guddo put her stick at the exact geometric centre. (a) Whose top will not spin at all? (b) Whose top will spin the best? Explain why.
[3 Marks]
✅ Step-by-Step Solution & Working
(a) Top that will not spin at all:
Zakir's top will not spin at all (or will wobble and instantly fall over) because placing the stick at the outer edge creates an extreme weight imbalance.
(b) Top that will spin the best:
Guddo's top will spin the best.
• Reason: When the pivot stick is placed at the exact center of the circle, the radius is equal in all directions. This distributes mass uniformly and balances the gravitational forces evenly around the axis of rotation, allowing the top to spin smoothly and stably for a long time.
Zakir's top will not spin at all (or will wobble and instantly fall over) because placing the stick at the outer edge creates an extreme weight imbalance.
(b) Top that will spin the best:
Guddo's top will spin the best.
• Reason: When the pivot stick is placed at the exact center of the circle, the radius is equal in all directions. This distributes mass uniformly and balances the gravitational forces evenly around the axis of rotation, allowing the top to spin smoothly and stably for a long time.
CBSE Step-by-Step Marking Scheme
Identifies Zakir's top as failing
1.0 M
Identifies Guddo's top as best with center balance explanation
2.0 M
Q3
Circle Geometry • Radius & Diameter
If the radius of a bicycle wheel is 35 cm, what is its diameter? If the diameter of a giant Ferris wheel is 24 metres, what is its radius?
[2 Marks]
✅ Step-by-Step Solution & Working
Formula: Diameter = 2 × Radius | Radius = Diameter ÷ 2
1. Bicycle Wheel:
Radius = 35 cm
Diameter = 2 × 35 cm = 70 cm.
2. Giant Ferris Wheel:
Diameter = 24 m
Radius = 24 m ÷ 2 = 12 metres.
1. Bicycle Wheel:
Radius = 35 cm
Diameter = 2 × 35 cm = 70 cm.
2. Giant Ferris Wheel:
Diameter = 24 m
Radius = 24 m ÷ 2 = 12 metres.
CBSE Step-by-Step Marking Scheme
Bicycle diameter: 70 cm
1.0 M
Ferris wheel radius: 12 m
1.0 M
Q4
Compass Construction • Drawing Circles
Describe how to draw a circle of radius 4 cm using a ruler and a geometric compass. What is the role of the fixed sharp point?
[3 Marks]
✅ Step-by-Step Solution & Working
Steps to Draw a Circle:
1. Setting the Radius: Open the arms of the compass. Place the sharp metal pointer at 0 cm mark on the ruler and expand the pencil tip to exactly the 4 cm mark.
2. Marking the Centre: Press the sharp metal pointer firmly onto paper without shifting; this point is the Centre (O) of the circle.
3. Drawing: Holding the compass strictly by the top handle, rotate the pencil 360° smoothly around the centre to trace the circular boundary.
The fixed sharp point ensures that every point on the circumference remains at an exact constant distance of 4 cm from the centre.
1. Setting the Radius: Open the arms of the compass. Place the sharp metal pointer at 0 cm mark on the ruler and expand the pencil tip to exactly the 4 cm mark.
2. Marking the Centre: Press the sharp metal pointer firmly onto paper without shifting; this point is the Centre (O) of the circle.
3. Drawing: Holding the compass strictly by the top handle, rotate the pencil 360° smoothly around the centre to trace the circular boundary.
The fixed sharp point ensures that every point on the circumference remains at an exact constant distance of 4 cm from the centre.
CBSE Step-by-Step Marking Scheme
Setting 4 cm measurement on ruler
1.0 M
Center anchor and 360° pencil rotation
1.0 M
Explaining constant distance from center
1.0 M
Q5
Building with Bricks • Brick Geometry & Measurement
A standard brick has length 20 cm, breadth 10 cm, and height 7 cm.
(a) What is the shape of a brick?
(b) What is the area of its largest face?
(c) How many bricks of 20 cm length placed end-to-end make a line 1 metre (100 cm) long?
[3 Marks]
(a) What is the shape of a brick?
(b) What is the area of its largest face?
(c) How many bricks of 20 cm length placed end-to-end make a line 1 metre (100 cm) long?
✅ Step-by-Step Solution & Working
Brick Mathematics:
• (a) Shape: A standard brick is a cuboid.
• (b) Area of largest face: Length × Breadth = 20 cm × 10 cm = 200 square centimetres (cm²).
• (c) Number of bricks in 1 m: 1 m = 100 cm. Number of bricks = 100 ÷ 20 = 5 bricks.
• (a) Shape: A standard brick is a cuboid.
• (b) Area of largest face: Length × Breadth = 20 cm × 10 cm = 200 square centimetres (cm²).
• (c) Number of bricks in 1 m: 1 m = 100 cm. Number of bricks = 100 ÷ 20 = 5 bricks.
CBSE Step-by-Step Marking Scheme
Cuboid shape
1 M
Area = 200 cm²
1 M
5 bricks
1 M
Q6
A Trip to Bhopal • Diesel Fuel Budget
During a school trip to Bhopal, 4 buses stop at a petrol pump to fill diesel. Each bus takes 50 litres of diesel. If 1 litre of diesel costs ₹70:
(a) How many litres of diesel are filled in total?
(b) What is the total bill amount the school must pay?
[3 Marks]
(a) How many litres of diesel are filled in total?
(b) What is the total bill amount the school must pay?
✅ Step-by-Step Solution & Working
Step-by-Step Solution:
• (a) Total diesel: 4 buses × 50 L = 200 litres.
• (b) Total bill amount: 200 L × ₹70/L = ₹14,000.
• (a) Total diesel: 4 buses × 50 L = 200 litres.
• (b) Total bill amount: 200 L × ₹70/L = ₹14,000.
CBSE Step-by-Step Marking Scheme
200 litres
1.5 M
₹14,000 bill
1.5 M
Q7
Long and Short • Height Comparisons
The world record for Men's High Jump is 245 cm and the Long Jump world record is 895 cm.
(a) Express the High Jump record in metres and centimetres.
(b) How many more centimetres are needed for the high jump record to reach exactly 2.5 metres (250 cm)?
[2 Marks]
(a) Express the High Jump record in metres and centimetres.
(b) How many more centimetres are needed for the high jump record to reach exactly 2.5 metres (250 cm)?
✅ Step-by-Step Solution & Working
Calculations:
• (a) High Jump: 245 cm = 2 metres 45 centimetres.
• (b) Difference: 250 cm - 245 cm = 5 centimetres.
• (a) High Jump: 245 cm = 2 metres 45 centimetres.
• (b) Difference: 250 cm - 245 cm = 5 centimetres.
CBSE Step-by-Step Marking Scheme
2 m 45 cm
1 M
5 cm difference
1 M
Q8
Tick-Tick-Tick • Expiry Dates on Food
A packet of butter shows 'Manufactured: 15/01/26' and 'Best before 180 days from manufacture'. Can it be consumed safely on 15th April 2026? Explain.
[2 Marks]
✅ Step-by-Step Solution & Working
Date Verification:
• From 15th Jan to 15th April is exactly 3 months (approx. 90 days).
• Since 90 days is well within the 180-day shelf life, the butter is completely safe to consume on 15th April 2026.
• From 15th Jan to 15th April is exactly 3 months (approx. 90 days).
• Since 90 days is well within the 180-day shelf life, the butter is completely safe to consume on 15th April 2026.
CBSE Step-by-Step Marking Scheme
Time elapsed is ~90 days
1 M
Safe to consume within 180 days
1 M
Q9
The Junk Seller • Loan and Interest
Kiran took a loan of ₹300 from a bank to set up her scrap shop. She pays back ₹51 every month for six months.
(a) How much money does she pay back in total?
(b) How much extra money (interest) did she pay to the bank?
[3 Marks]
(a) How much money does she pay back in total?
(b) How much extra money (interest) did she pay to the bank?
✅ Step-by-Step Solution & Working
Loan Repayment:
• (a) Total paid: 6 months × ₹51/month = ₹306.
• (b) Extra money (interest): ₹306 - ₹300 = ₹6.
• (a) Total paid: 6 months × ₹51/month = ₹306.
• (b) Extra money (interest): ₹306 - ₹300 = ₹6.
CBSE Step-by-Step Marking Scheme
Total repayment = ₹306
1.5 M
Interest = ₹6
1.5 M
Q10
Jugs and Mugs • Measuring Medicine Dose
A doctor prescribes a cough syrup bottle of 150 mL for a child. The child takes a 5 mL dose three times a day.
(a) How much cough syrup does the child take in one day?
(b) For how many days will the 150 mL bottle last?
[3 Marks]
(a) How much cough syrup does the child take in one day?
(b) For how many days will the 150 mL bottle last?
✅ Step-by-Step Solution & Working
Dosage Calculations:
• (a) Daily intake: 3 doses × 5 mL = 15 mL per day.
• (b) Days bottle lasts: 150 mL ÷ 15 mL/day = 10 days.
• (a) Daily intake: 3 doses × 5 mL = 15 mL per day.
• (b) Days bottle lasts: 150 mL ÷ 15 mL/day = 10 days.
CBSE Step-by-Step Marking Scheme
15 mL daily
1.5 M
Bottle lasts 10 days
1.5 M
Q11
Carts and Wheels • Circle Geometry
A cyclist has a bicycle wheel with a radius of 35 cm.
(a) What is the diameter of the wheel?
(b) What is the relationship between the radius and diameter of any circle?
[2 Marks]
(a) What is the diameter of the wheel?
(b) What is the relationship between the radius and diameter of any circle?
✅ Step-by-Step Solution & Working
Circle Properties:
• (a) Diameter: Diameter = 2 × Radius = 2 × 35 cm = 70 cm.
• (b) Relationship: The diameter is always twice the length of the radius (D = 2r), passing straight through the circle's center.
• (a) Diameter: Diameter = 2 × Radius = 2 × 35 cm = 70 cm.
• (b) Relationship: The diameter is always twice the length of the radius (D = 2r), passing straight through the circle's center.
CBSE Step-by-Step Marking Scheme
Diameter = 70 cm
1 M
Formula D = 2r stated
1 M
Q12
Halves and Quarters • Fraction Sharing
Raju has ₹100. He spends 1/2 of his money on a book, and 1/4 of his total money on a pen.
(a) How much does the book cost?
(b) How much does the pen cost?
(c) How much money does Raju have left?
[3 Marks]
(a) How much does the book cost?
(b) How much does the pen cost?
(c) How much money does Raju have left?
✅ Step-by-Step Solution & Working
Fractional Expenditure:
• (a) Book (1/2): ₹100 ÷ 2 = ₹50.
• (b) Pen (1/4): ₹100 ÷ 4 = ₹25.
• (c) Money Left: ₹100 - (₹50 + ₹25) = ₹100 - ₹75 = ₹25.
• (a) Book (1/2): ₹100 ÷ 2 = ₹50.
• (b) Pen (1/4): ₹100 ÷ 4 = ₹25.
• (c) Money Left: ₹100 - (₹50 + ₹25) = ₹100 - ₹75 = ₹25.
CBSE Step-by-Step Marking Scheme
Book = ₹50
1 M
Pen = ₹25
1 M
Remaining = ₹25
1 M
Q13
Play with Patterns • Secret Message Cipher
In a secret code, each letter is replaced by its position in the alphabet (A=1, B=2, C=3, ... Z=26).
Decode the secret message: '13 - 1 - 20 - 8 - 19'. What word does it spell?
[2 Marks]
Decode the secret message: '13 - 1 - 20 - 8 - 19'. What word does it spell?
✅ Step-by-Step Solution & Working
Decoding Cipher:
• 13 = M
• 1 = A
• 20 = T
• 8 = H
• 19 = S
The secret message spells MATHS.
• 13 = M
• 1 = A
• 20 = T
• 8 = H
• 19 = S
The secret message spells MATHS.
CBSE Step-by-Step Marking Scheme
Correct decoding to spell MATHS
2 M
Q14
Tables and Shares • Division with Remainder
A florist has 58 roses. She wants to make bouquets containing 6 roses each.
(a) How many complete bouquets can she make?
(b) How many roses will be left over?
[2 Marks]
(a) How many complete bouquets can she make?
(b) How many roses will be left over?
✅ Step-by-Step Solution & Working
Division:
• 58 ÷ 6 = 9 with a remainder of 4.
• (a) Complete bouquets: 9 bouquets (9 × 6 = 54 roses).
• (b) Leftover roses: 58 - 54 = 4 roses left over.
• 58 ÷ 6 = 9 with a remainder of 4.
• (a) Complete bouquets: 9 bouquets (9 × 6 = 54 roses).
• (b) Leftover roses: 58 - 54 = 4 roses left over.
CBSE Step-by-Step Marking Scheme
9 bouquets
1 M
4 leftover roses
1 M
Q15
Review Questions • True or False
State True or False:
(a) A square has 4 equal sides and its perimeter is 4 × side.
(b) 1 kilometre equals 100 metres.
(c) Half past 4 in the afternoon is written as 16:30 in 24-hour time.
[3 Marks]
(a) A square has 4 equal sides and its perimeter is 4 × side.
(b) 1 kilometre equals 100 metres.
(c) Half past 4 in the afternoon is written as 16:30 in 24-hour time.
✅ Step-by-Step Solution & Working
Answers:
• (a) True: Perimeter = 4 × side.
• (b) False: 1 km = 1000 metres.
• (c) True: 4:30 PM = 12:00 + 4:30 = 16:30 hours.
• (a) True: Perimeter = 4 × side.
• (b) False: 1 km = 1000 metres.
• (c) True: 4:30 PM = 12:00 + 4:30 = 16:30 hours.
CBSE Step-by-Step Marking Scheme
True, False, True with explanations
3 M
🌟 NEP 2020 Competency-Based Case Study
Playground Circle Marker with String
The sports teacher needs to draw a circular kabaddi ring on the ground with a diameter of 8 metres. She has a wooden peg and a coil of rope.
Analytical Task: How long should she cut the rope to tie between the central peg and the chalk stick?
Evaluation & Model Answer:
Answer: 4 metres.
Explanation: The length of the rope between the central fixed peg (centre) and the rotating chalk stick represents the radius of the circle.
Radius = Diameter ÷ 2 = 8 m ÷ 2 = 4 metres.
Explanation: The length of the rope between the central fixed peg (centre) and the rotating chalk stick represents the radius of the circle.
Radius = Diameter ÷ 2 = 8 m ÷ 2 = 4 metres.
Rubric: 1 Mark for identifying rope as radius; 1 Mark for calculating 4 metres.