Chapter 1 of 14
Chapter 1: Shapes Around Us
Exploring 2D & 3D Shapes, Faces, Edges, Vertices, Tangrams & Tessellations | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- Distinction between 2D flat shapes (squares, rectangles, circles, triangles) and 3D solid shapes (cubes, cuboids, cylinders, cones, spheres).
- Counting flat faces, curved faces, straight edges, curved edges, and vertices on everyday objects.
- Understanding rolling vs. sliding properties based on curved and flat surfaces.
- Creating pictures and figures using the 7-piece Tangram puzzle.
- Tessellation: Tiling patterns that cover a plane completely without any gaps or overlaps.
All Textbook Exercises & In-Text Tasks (15 Questions)
100% Complete Ingestion
Q1
Textbook Exercise • 3D Properties
Look around your classroom and home. Name two objects each that: (a) Only roll, (b) Only slide, (c) Both roll and slide. Explain the geometric reason for each behavior.
[3 Marks]
✅ Step-by-Step Solution & Working
(a) Objects that only roll:
• Examples: A cricket ball, a marble, an orange.
• Reason: These objects have only a continuous curved surface and no flat faces or straight edges, so they rotate smoothly over surfaces.
(b) Objects that only slide:
• Examples: A textbook, a rectangular pencil box, an eraser.
• Reason: These objects have only flat planar faces and straight edges. When pushed, a flat face stays in contact with the table and slides without rolling.
(c) Objects that both roll and slide:
• Examples: A one-rupee coin, a water bottle, a round cylindrical marker pen.
• Reason: These objects possess both flat faces and a curved surface. When placed on their flat circular base, they slide; when placed on their curved lateral side, they roll.
• Examples: A cricket ball, a marble, an orange.
• Reason: These objects have only a continuous curved surface and no flat faces or straight edges, so they rotate smoothly over surfaces.
(b) Objects that only slide:
• Examples: A textbook, a rectangular pencil box, an eraser.
• Reason: These objects have only flat planar faces and straight edges. When pushed, a flat face stays in contact with the table and slides without rolling.
(c) Objects that both roll and slide:
• Examples: A one-rupee coin, a water bottle, a round cylindrical marker pen.
• Reason: These objects possess both flat faces and a curved surface. When placed on their flat circular base, they slide; when placed on their curved lateral side, they roll.
CBSE Step-by-Step Marking Scheme
Correct examples for all 3 categories
1.5 M
Scientific explanation relating flat/curved faces to rolling/sliding
1.5 M
Q2
Textbook Exercise • Geometric Attributes
Complete the table by counting the number of faces, edges, and corners (vertices) for: (1) A chalk box (Cube), (2) A matchbox (Cuboid), (3) A new unsharpened pencil (Hexagonal Prism / Cylinder), (4) A football (Sphere).
[4 Marks]
✅ Step-by-Step Solution & Working
| Shape / Object | Flat Faces | Curved Faces | Total Edges | Corners (Vertices) |
|---|---|---|---|---|
| 1. Chalk box (Cube) | 6 square faces | 0 | 12 straight edges | 8 corners |
| 2. Matchbox (Cuboid) | 6 rectangular faces | 0 | 12 straight edges | 8 corners |
| 3. Cylinder (Can/Pencil) | 2 circular faces | 1 curved face | 2 curved edges | 0 corners |
| 4. Football (Sphere) | 0 | 1 curved face | 0 edges | 0 corners |
CBSE Step-by-Step Marking Scheme
Correct counts for Cube & Cuboid
2.0 M
Correct counts for Cylinder & Sphere
2.0 M
Q3
Tangram & Tessellation Investigation
What is a Tangram? How many pieces does a standard Chinese tangram have, and which geometric shapes are present in it? Can a circle tessellate on a floor? Why or why not?
[3 Marks]
✅ Step-by-Step Solution & Working
1. Tangram Definition & Composition:
A Tangram is a traditional geometric dissection puzzle consisting of a square cut into 7 flat pieces called 'tans':
• 5 Triangles: 2 large right-angled isosceles triangles, 1 medium right-angled triangle, and 2 small right-angled triangles.
• 1 Square: Formed using 2 small triangles.
• 1 Parallelogram: Formed with opposite sides equal and parallel.
2. Can Circles Tessellate?
• Answer: No, circles cannot tessellate.
• Mathematical Reason: Tessellation requires shapes to fit together leaving zero gaps and zero overlaps. When round circular coins or tiles are placed side by side, empty curved spaces (interstices) are always left between them. Therefore, circles cannot tile a flat surface completely.
A Tangram is a traditional geometric dissection puzzle consisting of a square cut into 7 flat pieces called 'tans':
• 5 Triangles: 2 large right-angled isosceles triangles, 1 medium right-angled triangle, and 2 small right-angled triangles.
• 1 Square: Formed using 2 small triangles.
• 1 Parallelogram: Formed with opposite sides equal and parallel.
2. Can Circles Tessellate?
• Answer: No, circles cannot tessellate.
• Mathematical Reason: Tessellation requires shapes to fit together leaving zero gaps and zero overlaps. When round circular coins or tiles are placed side by side, empty curved spaces (interstices) are always left between them. Therefore, circles cannot tile a flat surface completely.
CBSE Step-by-Step Marking Scheme
Correct description of 7 tangram pieces
1.5 M
Logical reason why circles leave gaps and cannot tessellate
1.5 M
Q4
3D Shapes • Nets & Vertices
How many faces, edges, and vertices does a cuboid have? Draw or describe the net of an open rectangular box.
[3 Marks]
✅ Step-by-Step Solution & Working
1. Properties of a Cuboid:
• Faces: 6 flat rectangular faces (opposite faces are identical in size).
• Edges: 12 straight edges where two faces meet.
• Vertices (Corners): 8 vertices where three edges meet.
2. Net of an Open Box (without lid):
An open box has 5 faces (1 base + 4 side walls). Its net consists of a central rectangular base surrounded by four rectangular flaps attached to its four sides, forming a cross shape (+). When the four flaps are folded upward at 90°, they form the four walls of the open box.
• Faces: 6 flat rectangular faces (opposite faces are identical in size).
• Edges: 12 straight edges where two faces meet.
• Vertices (Corners): 8 vertices where three edges meet.
2. Net of an Open Box (without lid):
An open box has 5 faces (1 base + 4 side walls). Its net consists of a central rectangular base surrounded by four rectangular flaps attached to its four sides, forming a cross shape (+). When the four flaps are folded upward at 90°, they form the four walls of the open box.
CBSE Step-by-Step Marking Scheme
Correct counts: 6 faces, 12 edges, 8 vertices
1.5 M
Accurate structural description of 5-face cross net
1.5 M
Q5
Textbook Exercise • Real-World Context
In Chapter 1 ('Shapes Around Us'), a classroom project investigates real-world mathematics: If a school bus travels 48 kilometres on 4 litres of diesel, what is the mileage (kilometres per litre) of the bus? How much diesel is needed for a 120-kilometre journey?
[3 Marks]
✅ Step-by-Step Solution & Working
Step-by-Step Solution:
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
CBSE Step-by-Step Marking Scheme
Mileage = 12 km/L
1.5 M
Diesel needed = 10 L
1.5 M
Q6
Practice Time • Measurement & Conversions
Convert the following measurements into standard units:
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
[3 Marks]
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
✅ Step-by-Step Solution & Working
Conversions:
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
CBSE Step-by-Step Marking Scheme
465 cm
1 M
3 L 500 mL
1 M
2750 g
1 M
Q7
Try These • Geometry and Shapes
Look at a solid brick:
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
[3 Marks]
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
✅ Step-by-Step Solution & Working
Properties of a Brick (Cuboid):
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
CBSE Step-by-Step Marking Scheme
6 faces
1 M
Rectangular faces
1 M
12 edges and 8 corners
1 M
Q8
Money & Accounts • Rate List Calculation
At a recycling shop, the rate list shows: Waste paper = ₹12 per kg, Plastic = ₹18 per kg, Iron = ₹25 per kg.
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
[3 Marks]
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
✅ Step-by-Step Solution & Working
Calculations:
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
CBSE Step-by-Step Marking Scheme
Individual costs (₹120, ₹90, ₹100)
1.5 M
Total = ₹310
1.5 M
Q9
Fractions Exploration • Parts of a Whole
A rectangular chocolate bar has 16 equal pieces. Reena gives 1/4 of the chocolate to her brother and 1/2 of it to her friend.
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
[3 Marks]
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
✅ Step-by-Step Solution & Working
Fractional Breakdown:
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
CBSE Step-by-Step Marking Scheme
Brother = 4 pieces
1 M
Friend = 8 pieces
1 M
Reena = 4 pieces
1 M
Q10
Time & Clocks • Elapsed Time
A school cricket match began at 9:30 AM and concluded at 1:15 PM.
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
[3 Marks]
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
✅ Step-by-Step Solution & Working
Time Calculations:
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
CBSE Step-by-Step Marking Scheme
13:15 hours in 24-hr clock
1 M
Duration = 3 hours 45 minutes with steps
2 M
Q11
Perimeter & Area • Fencing a Field
A square vegetable garden has a side length of 15 metres.
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
[3 Marks]
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
✅ Step-by-Step Solution & Working
Perimeter & Cost:
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
CBSE Step-by-Step Marking Scheme
Perimeter = 60 m
1.5 M
Cost = ₹1,200
1.5 M
Q12
Mental Math • Division and Multiplication
Solve mentally using number breakdown:
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
[2 Marks]
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
✅ Step-by-Step Solution & Working
Solutions:
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
CBSE Step-by-Step Marking Scheme
All 4 mental math facts correct
2 M
Q13
Patterns & Symmetries • Rotational Turns
Look at the letter 'H' and the letter 'N':
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
[2 Marks]
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
✅ Step-by-Step Solution & Working
Rotational Symmetry:
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
CBSE Step-by-Step Marking Scheme
Half-turn explanation (same)
1 M
Quarter-turn explanation
1 M
Q14
Smart Charts • Data Interpretation
In a Class 4 survey of 40 children, 20 like Cricket, 10 like Football, and 10 like Badminton.
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
[3 Marks]
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
✅ Step-by-Step Solution & Working
Data & Chapati Chart:
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
CBSE Step-by-Step Marking Scheme
Fraction = 1/2 for cricket
1.5 M
Pie chart sectors explained (1/2, 1/4, 1/4)
1.5 M
Q15
Chapter Review • True or False
State True or False:
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
[3 Marks]
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
✅ Step-by-Step Solution & Working
Answers:
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
CBSE Step-by-Step Marking Scheme
True, False, True with explanations
3 M
🌟 NEP 2020 Competency-Based Case Study
Floor Tiling Design for the School Assembly Hall
The school headmaster wants to tile the newly renovated assembly floor. The mason proposes two tile options: Hexagonal tiles and Pentagonal tiles. Rahul observes that honeycombs in nature use hexagons without any gaps.
Analytical Task: Which of the two shapes (regular hexagon or regular pentagon) will successfully tile the floor without leaving empty gaps? State the geometric reason.
Evaluation & Model Answer:
Selected Shape: Regular Hexagon.
Reason: The interior angle of a regular hexagon is 120°. When three hexagons meet at a single vertex point, their angles sum to exactly 120° + 120° + 120° = 360°, which forms a complete flat turn leaving zero gaps. In contrast, regular pentagons have interior angles of 108°; three pentagons sum to 324° (leaving a 36° gap) and four pentagons sum to 432° (overlapping). Hence, between the two proposed options, only the regular hexagon can tile the floor. (Note: In plane geometry, the only regular polygons that can tile a flat surface without gaps are equilateral triangles, squares, and regular hexagons; regular pentagons cannot).
Reason: The interior angle of a regular hexagon is 120°. When three hexagons meet at a single vertex point, their angles sum to exactly 120° + 120° + 120° = 360°, which forms a complete flat turn leaving zero gaps. In contrast, regular pentagons have interior angles of 108°; three pentagons sum to 324° (leaving a 36° gap) and four pentagons sum to 432° (overlapping). Hence, between the two proposed options, only the regular hexagon can tile the floor. (Note: In plane geometry, the only regular polygons that can tile a flat surface without gaps are equilateral triangles, squares, and regular hexagons; regular pentagons cannot).
Rubric: 1 Mark for choosing hexagon; 1 Mark for explaining angle sum of 360° at vertex.