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Visualising Solid Shapes – CBSE NCERT Study Resources

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7th

7th - Mathematics

Visualising Solid Shapes

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Chapter Overview: Visualising Solid Shapes

This chapter introduces students to the concept of solid shapes and their visualisation in three-dimensional space. It covers various types of solids, their properties, and methods to represent them on a two-dimensional plane. The chapter also explores nets of solids and different views of 3D objects.

Solid Shapes: Three-dimensional objects that occupy space and have length, breadth, and height.

Types of Solid Shapes

The chapter discusses the following common solid shapes:

  • Cubes
  • Cuboids
  • Pyramids
  • Prisms
  • Cylinders
  • Cones
  • Spheres

Faces, Edges, and Vertices: The flat surfaces of a solid are called faces, the line segments where faces meet are edges, and the corners where edges meet are vertices.

Visualising 3D Shapes in 2D

The chapter explains different methods to represent 3D shapes on paper:

  • Oblique sketches
  • Isometric sketches
  • Orthographic projections

Nets of Solid Shapes

Net: A two-dimensional shape that can be folded to form a three-dimensional solid.

Students learn to identify and draw nets for various solids like cubes, cuboids, pyramids, and prisms.

Viewing Different Perspectives

The chapter covers how objects appear different when viewed from various angles:

  • Front view
  • Side view
  • Top view

Mapping Space Around Us

Students learn to understand and represent the spatial arrangement of objects through:

  • Plans and elevations
  • Scale drawings
  • Shadow play of 3D objects

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
Name the solid shape with 6 square faces.
Answer:
Cube
Question 2:
How many edges does a triangular prism have?
Answer:
9 edges
Question 3:
Identify the 3D shape formed by rotating a rectangle about one side.
Answer:
Cylinder
Question 4:
What is the net of a cube?
Answer:

6 squares joined edge-to-edge.

Question 5:
Name the solid with one circular base and one vertex.
Answer:
Cone
Question 6:
How many vertices does a square pyramid have?
Answer:
5 vertices
Question 7:
Which 3D shape has all points equidistant from its center?
Answer:
Sphere
Question 8:
What is the cross-section of a cube parallel to its base?
Answer:
Square
Question 9:
Name the solid with two triangular and three rectangular faces.
Answer:
Triangular prism
Question 10:
How many faces does a rectangular pyramid have?
Answer:
5 faces
Question 11:
Which net forms a cylinder?
Answer:

Rectangle + two circles.

Question 12:
What is the shape of a football?
Answer:
Sphere
Question 13:
What is the difference between a 2D shape and a 3D shape?
Answer:

A 2D shape has only length and width (like a square or circle), while a 3D shape has length, width, and height (like a cube or sphere).
2D shapes are flat, whereas 3D shapes are solid and occupy space.

Question 14:
Name the three-dimensional shape that has 6 square faces.
Answer:

The shape is a cube.
All its faces are equal squares, and it has 12 edges and 8 vertices.

Question 15:
What is the net of a 3D shape?
Answer:

A net is a 2D layout that can be folded to form a 3D shape.
For example, the net of a cube consists of 6 squares arranged in a cross pattern.

Question 16:
Identify the solid shape that has one curved surface and no vertices.
Answer:

The shape is a sphere.
It has no edges or vertices, just a smooth curved surface.

Question 17:
How many faces does a square pyramid have?
Answer:

A square pyramid has 5 faces.
It has 1 square base and 4 triangular lateral faces.

Question 18:
What is the shape of the base of a cylinder?
Answer:

The base of a cylinder is a circle.
It has two circular bases and one curved surface.

Question 19:
Name the 3D shape formed by rotating a rectangle about one of its sides.
Answer:

The shape formed is a cylinder.
Rotating a rectangle along its length or width creates this solid.

Question 20:
How many vertices does a rectangular prism have?
Answer:

A rectangular prism has 8 vertices.
It has 6 rectangular faces and 12 edges.

Question 21:
What is the Euler's formula for polyhedrons?
Answer:

Euler's formula states:
Faces (F) + Vertices (V) - Edges (E) = 2.
For example, a cube has 6 faces, 8 vertices, and 12 edges: 6 + 8 - 12 = 2.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
Identify the number of faces, edges, and vertices in a cube.
Answer:

A cube has:
6 faces (all square-shaped),
12 edges (where two faces meet),
8 vertices (corners where edges meet).

Question 2:
What is the difference between a prism and a pyramid?
Answer:

A prism has two identical parallel bases and rectangular faces.
A pyramid has one base and triangular faces meeting at a common vertex.

Question 3:
Name the solid shape that has no edges and no vertices.
Answer:

A sphere has no edges or vertices. It is a perfectly round 3D shape.

Question 4:
Which 3D shape has all faces as squares?
Answer:

A cube has all its faces as squares. All edges are of equal length.

Question 5:
Can a polyhedron have 4 triangular faces? If yes, name it.
Answer:

Yes, a triangular pyramid (or tetrahedron) has 4 triangular faces.

Question 6:
What is the net of a cube?
Answer:

The net of a cube is a 2D arrangement of 6 squares connected edge-to-edge that can fold into a cube.

Question 7:
How many vertices does a rectangular pyramid have?
Answer:

A rectangular pyramid has 5 vertices:
4 at the base corners,
1 at the apex (top).

Question 8:
Name a solid shape that has one curved surface and one circular face.
Answer:

A cone has one curved surface and one circular face as its base.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
Identify the number of faces, edges, and vertices in a cube. Draw a labelled diagram to support your answer.
Answer:

A cube is a three-dimensional shape with equal sides. Here are its properties:


Faces: 6 (all square-shaped)
Edges: 12 (where two faces meet)
Vertices: 8 (corners where edges meet)

Diagram:


[Draw a cube with labels: Faces (F), Edges (E), Vertices (V)]
Question 2:
Differentiate between a prism and a pyramid with an example of each.
Answer:

Prism: A prism has two identical parallel bases and rectangular lateral faces.
Example: Triangular prism (bases are triangles).


Pyramid: A pyramid has one base and triangular lateral faces meeting at a apex.
Example: Square pyramid (base is a square).

Question 3:
Explain how a net helps in visualising a 3D shape. Give an example of a net for a cylinder.
Answer:

A net is a 2D layout that folds into a 3D shape. It helps understand the structure by showing all faces.


Example: Cylinder net consists of:

  • One rectangle (lateral surface)
  • Two circles (top and bottom bases)

[Draw the net with labels]
Question 4:
A solid has 5 faces and 9 edges. Identify the shape and verify using Euler’s formula.
Answer:

The shape is a square pyramid (4 triangular faces + 1 square base).


Verification using Euler’s formula (F + V - E = 2):


Faces (F) = 5
Edges (E) = 9
Vertices (V) = 5 + 2 - 9 = 6
5 + 6 - 9 = 2 ✔️
Question 5:
Describe how a tetrahedron differs from a triangular prism in terms of faces and edges.
Answer:

Tetrahedron: 4 triangular faces, 6 edges (all faces are triangles).


Triangular prism: 2 triangular bases + 3 rectangular faces, 9 edges.


Key difference: A tetrahedron is a pyramid, while a triangular prism has parallel bases.

Question 6:
Identify the number of faces, edges, and vertices in a cube. Draw its net.
Answer:

A cube is a three-dimensional shape with equal sides. Here are its properties:


Faces: 6 (all square-shaped)
Edges: 12 (where two faces meet)
Vertices: 8 (corners where edges meet)

For the net, draw six squares arranged in a cross or any other valid 2D layout that can fold into a cube.

Question 7:
Differentiate between a prism and a pyramid with examples.
Answer:

Prism: A prism has two identical polygonal bases and rectangular lateral faces. Example: Triangular prism (base: triangle).
Pyramid: A pyramid has one polygonal base and triangular lateral faces meeting at a common vertex. Example: Square pyramid (base: square).

Key difference: Prisms have two bases, while pyramids have only one.

Question 8:
Explain how a cylinder is different from a cone in terms of faces and vertices.
Answer:

Cylinder: It has two circular faces (top and bottom) and one curved face. It has no vertices.
Cone: It has one circular face (base) and one curved face tapering to a vertex (apex). It has one vertex.

Note: Both are circular in shape, but a cone narrows to a point.

Question 9:
What is an oblique sketch? How is it useful in visualising solid shapes?
Answer:

An oblique sketch is a 2D drawing of a 3D shape where the front face is drawn in true shape, and the side faces are drawn at an angle (usually 45°).


Usefulness:

  • Helps visualize depth and dimensions of solids.
  • Simplifies complex shapes for better understanding.
  • Used in engineering and design sketches.
Question 10:
Draw the net of a triangular prism and label its dimensions.
Answer:

The net of a triangular prism consists of:


1. Two identical triangles (bases).
2. Three rectangles (lateral faces connecting the bases).

Label the sides as follows:

  • Triangles: Base = b, Height = h.
  • Rectangles: Length = l (prism height), Width = sides of the triangle.

Arrange them in a sequence where rectangles join the corresponding sides of the triangles.

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Explain the Euler’s formula for polyhedrons with an example. How is it useful in visualising solid shapes?
Answer:
Introduction

We studied that Euler’s formula relates the number of vertices (V), edges (E), and faces (F) of a polyhedron. It is given by V - E + F = 2.


Argument 1

For a cube, V = 8, E = 12, and F = 6. Applying Euler’s formula: 8 - 12 + 6 = 2, which holds true.


Argument 2

Our textbook shows that this formula helps verify if a given net can form a closed polyhedron. It ensures the shape is valid.


Conclusion

Euler’s formula is essential for analysing and constructing polyhedrons in real-life, like packaging boxes.

Question 2:
Differentiate between prisms and pyramids with examples. How are their nets different?
Answer:
Introduction

Prisms and pyramids are 3D shapes, but their structures differ. A prism has two identical bases, while a pyramid has one base and triangular faces.


Argument 1

A triangular prism has two triangular bases and three rectangular faces. Its net includes two triangles and three rectangles.


Argument 2

A square pyramid has one square base and four triangular faces. Its net shows one square and four triangles joined at the apex.


Conclusion

Understanding nets helps us visualise how these shapes unfold, like in origami or packaging.

Question 3:
Describe how to identify the front, top, and side views of a 3D shape. Use a cylinder as an example.
Answer:
Introduction

We learned that 3D shapes can be viewed from the front, top, and side to understand their dimensions.


Argument 1

For a cylinder, the front view is a rectangle (height × length), while the side view is identical due to symmetry.


Argument 2

The top view is a circle, representing the circular base. Our textbook shows this using a water bottle as a real-life example.


Conclusion

These views help architects and designers sketch objects accurately.

Question 4:
Explain the Euler's formula for polyhedrons with an example. Verify it for a cube.
Answer:
Introduction

We studied that Euler's formula relates the number of faces (F), vertices (V), and edges (E) of a polyhedron as F + V - E = 2.


Argument 1

Our textbook shows a cube has 6 faces, 8 vertices, and 12 edges. Applying the formula: 6 + 8 - 12 = 2, which matches Euler's result.


Argument 2

Another example is a triangular prism (F=5, V=6, E=9). Here, 5 + 6 - 9 = 2, confirming the formula.


Conclusion

Euler's formula helps verify the structure of polyhedrons in real-life objects like boxes or pyramids.

Question 5:
Describe how to identify the nets of a cube. Draw any two valid nets.
Answer:
Introduction

We learned that a net is a 2D shape that folds into a 3D solid like a cube.


Argument 1

Our textbook shows 11 possible nets for a cube. One net has 6 squares in a row, with four squares attached to the middle square.


Argument 2

Another net is a 'T-shape' with three squares on top and bottom of a central square. [Diagram: T-shaped net]. Both nets fold into cubes without overlaps.


Conclusion

Nets help visualize packaging designs, like cardboard boxes.

Question 6:
Compare the properties of a prism and a pyramid with examples.
Answer:
Introduction

We studied that prisms and pyramids are polyhedrons but differ in structure.


Argument 1

A prism has two identical bases (e.g., triangular prism) and rectangular faces. Our textbook shows a rectangular prism with 6 faces and 12 edges.


Argument 2

A pyramid has one base and triangular faces meeting at a vertex. For example, a square pyramid has 5 faces and 8 edges.


Conclusion

Prisms are used in boxes, while pyramids appear in monuments like the Pyramids of Egypt.

Question 7:
Explain the Euler’s formula for polyhedrons with examples. How is it useful in identifying shapes?
Answer:
Introduction

We studied that Euler’s formula relates the number of faces (F), vertices (V), and edges (E) of a polyhedron as F + V - E = 2.


Argument 1
  • Example: A cube has 6 faces, 8 vertices, and 12 edges. Applying the formula: 6 + 8 - 12 = 2.
  • Our textbook shows a triangular prism with 5 faces, 6 vertices, and 9 edges: 5 + 6 - 9 = 2.

Conclusion

This formula helps verify if a given net can form a closed 3D shape.

Question 8:
Compare the properties of a cube and a cuboid using their faces, edges, and vertices.
Answer:
Introduction

Both cube and cuboid are polyhedrons, but their dimensions differ.


Argument 1
  • Cube: All faces are squares (6), edges (12), and vertices (8).
  • Cuboid: Faces are rectangles (6), edges (12), and vertices (8).

Argument 2

Our textbook shows a cube has equal edge lengths, while a cuboid’s edges vary.


Conclusion

Both share Euler’s formula but differ in face shapes and symmetry.

Question 9:
Describe how to identify the nets of a cylinder. Give one real-life example.
Answer:
Introduction

A net is a 2D shape that folds into a 3D figure like a cylinder.


Argument 1
  • Our textbook shows a cylinder’s net has two circles (bases) and one rectangle (lateral face).
  • Real-life example: A tin can’s label is the rectangle, and its lids are the circles.

Conclusion

Nets help visualize how 3D shapes are constructed from 2D parts.

Question 10:
Explain the Euler’s formula for polyhedrons with an example. Verify it for a cube.
Answer:
Introduction

We studied that Euler’s formula relates the number of vertices (V), edges (E), and faces (F) of a polyhedron. The formula is V - E + F = 2.


Argument 1

Our textbook shows a cube has 8 vertices, 12 edges, and 6 faces. Applying Euler’s formula: 8 - 12 + 6 = 2, which matches the formula.


Argument 2

Another example is a triangular prism with 6 vertices, 9 edges, and 5 faces. Here, 6 - 9 + 5 = 2, proving the formula.


Conclusion

Euler’s formula helps verify polyhedron structures, as seen in real-life objects like dice or boxes.

Question 11:
Differentiate between prisms and pyramids with examples. Include a real-life application for each.
Answer:
Introduction

We learned that prisms and pyramids are 3D shapes but differ in their base and faces.


Argument 1

A prism has two identical bases (e.g., triangular prism) and rectangular faces. A pyramid has one base (e.g., square pyramid) and triangular faces meeting at a vertex.


Argument 2

Real-life examples: Prisms are used in glass prisms for light dispersion, while pyramids are seen in the shape of tents or monuments.


Conclusion

Understanding these shapes helps us identify objects like buildings or optical tools.

Question 12:
Draw the 2D nets of a cylinder and a cone. Explain how these nets form the 3D shapes.
Answer:
Introduction

Nets are 2D layouts that fold into 3D shapes. We studied nets for cylinders and cones.


Argument 1

A cylinder’s net has two circles (bases) and a rectangle (curved surface). Folding the rectangle joins the circles to form the cylinder.


Argument 2

A cone’s net has a circle (base) and a sector (curved surface). Rolling the sector forms the cone’s pointed top.


Conclusion

Nets help visualize how 3D shapes like cans or party hats are constructed.

Question 13:
Explain the differences between prisms and pyramids with suitable examples and diagrams. How are their faces, edges, and vertices different?
Answer:

Both prisms and pyramids are three-dimensional shapes, but they have distinct features:


Prisms:

  • They have two identical parallel bases (e.g., triangles, rectangles).
  • The sides (lateral faces) are rectangles or parallelograms.
  • Example: A rectangular prism (like a box) has 6 faces, 12 edges, and 8 vertices.

Pyramids:

  • They have a single base (e.g., square, triangle) and triangular lateral faces meeting at a common apex.
  • Example: A square pyramid has 5 faces (1 square base + 4 triangular faces), 8 edges, and 5 vertices.

Key Differences:

  • Prisms have two bases; pyramids have one.
  • Prisms have rectangular lateral faces; pyramids have triangular ones.
  • Edges and vertices vary based on the base shape.

Diagrams would show a prism with equal top/bottom faces and a pyramid with a pointed top.

Question 14:
Describe how to identify the net of a cube. Draw at least two different nets and explain why they can form a cube.
Answer:

A net is a 2D shape that can be folded to form a 3D solid like a cube. To identify a cube net:


Rules:

  • It must have 6 squares (for the 6 faces of a cube).
  • Squares must be connected edge-to-edge (no overlapping or gaps).
  • When folded, all squares must meet to form a closed shape.

Example Nets:

  • Net 1: 4 squares in a row with 1 square attached above the second and 1 below the third square.
  • Net 2: 3 squares in a row with 1 square attached above the middle and 2 squares stacked to one side.

Why They Work: Both nets have 6 squares connected so that folding along edges brings all faces together to form a cube. Diagrams would show the nets with fold lines and how they assemble into a cube.

Question 15:
Explain the difference between a prism and a pyramid with suitable examples and diagrams. Also, mention the number of faces, edges, and vertices for each.
Answer:

A prism and a pyramid are both three-dimensional shapes, but they have distinct features.

Prism:
1. A prism has two identical parallel bases (like triangles, rectangles, etc.) connected by rectangular faces.
2. Example: A triangular prism has 5 faces (2 triangular bases + 3 rectangular lateral faces), 9 edges, and 6 vertices.
3. Diagram: Draw two congruent triangles (bases) connected by three rectangles.

Pyramid:
1. A pyramid has a single polygonal base and triangular faces meeting at a common vertex (apex).
2. Example: A square pyramid has 5 faces (1 square base + 4 triangular lateral faces), 8 edges, and 5 vertices.
3. Diagram: Draw a square base with four triangles converging at the apex.

Key Difference: A prism has two bases, while a pyramid has only one base.

Question 16:
Describe how to identify the net of a cube. Draw at least two different nets of a cube and explain why they can form a cube when folded.
Answer:

A net is a 2D shape that can be folded to form a 3D object like a cube. To identify a net of a cube:


  • It must have six square faces arranged such that they can fold into a closed cube.
  • No overlapping squares when folded, and all edges must align perfectly.

Example Nets:

  1. Cross-shaped net: One central square with four squares attached to its sides and one square attached to any outer square.
    When folded, the four side squares form the cube's sides, and the top/bottom squares close the shape.
  2. Staircase net: Three squares in a row, with one square attached to the top of the middle square and two squares stacked vertically beside it.
    Folding the "steps" creates the cube's structure.

Diagram: Sketch both nets with labels showing folding directions. Highlight that all nets must have 6 connected squares without gaps.

Question 17:
Explain the difference between a prism and a pyramid with suitable examples and diagrams. Also, mention the number of faces, edges, and vertices for each shape.
Answer:

A prism and a pyramid are both three-dimensional shapes, but they have distinct features. Here’s a detailed comparison:


Prism:

  • A prism has two identical parallel bases (e.g., triangles, rectangles).
  • The sides (lateral faces) are rectangles or parallelograms.
  • Example: A rectangular prism (like a box) has 6 faces, 12 edges, and 8 vertices.

Pyramid:

  • A pyramid has one base (e.g., square, triangle) and triangular lateral faces that meet at a common vertex (apex).
  • Example: A square pyramid has 5 faces (1 square base + 4 triangular faces), 8 edges, and 5 vertices.

Diagram: Draw a rectangular prism and a square pyramid side by side, labeling their faces, edges, and vertices.


Key Difference: A prism has two parallel bases, while a pyramid has one base and an apex.

Question 18:
Describe how a net of a 3D shape helps in visualizing and constructing it. Take the example of a cube and explain step-by-step how its net can be drawn and folded to form the shape.
Answer:

A net is a 2D layout of a 3D shape that can be folded to form the solid. It helps in visualizing the structure and understanding the arrangement of faces.


Example: Cube

A cube has 6 square faces. Its net can be drawn in multiple ways, but one common arrangement is a cross-shaped net with one central square and four squares attached to its sides, plus one square above or below.


Steps to draw and fold:


1. Draw a central square.
2. Attach four squares to its top, bottom, left, and right sides.
3. Place the sixth square either above or below one of the side squares.
4. Cut out the net and fold along the edges where squares meet.
5. The squares will form the six faces of the cube when folded.

This method ensures all edges align perfectly to create a closed 3D shape. Diagrams should show the net before folding and the final cube after folding.

Question 19:
Describe the process of visualizing a 3D shape from its 2D net. Use the example of a cube and explain how its net can be folded to form the shape. Draw the net and label its parts.
Answer:

Visualizing a 3D shape from its 2D net involves understanding how the flat layout folds to form the solid. Here’s how it works for a cube:


Step 1: Identify the Net
A cube has 6 square faces. Its net consists of 6 squares arranged in a way that they can be folded to form the cube. There are 11 possible nets for a cube.


Step 2: Draw the Net
Draw a common net: a central square with 4 squares attached to its sides (like a cross) and 1 square attached to any outer square.


Step 3: Folding Process
1. Fold the side squares upward to form the cube’s walls.
2. Fold the top square to cover the cube.
3. The bottom square becomes the base.


Diagram: Sketch the net with labels (e.g., "Top," "Front," "Base") and arrows showing folding directions.


Application: Nets help in understanding surface area calculations and real-life packaging designs.

Question 20:
Describe the process of visualizing a net of a cube. Draw the net and explain how folding it forms the cube. Also, list all possible unique nets for a cube.
Answer:

A net is a two-dimensional shape that can be folded to form a three-dimensional solid like a cube.

Process:
1. A cube has 6 square faces. A net must show all 6 squares connected edge-to-edge.
2. Example net: Draw 4 squares in a row (middle squares represent the front, back, left, and right faces). Attach one square above the second square (top face) and one below the third square (bottom face).
3. Folding: Fold the squares upward/downward along the edges to form the cube.

Unique Nets: There are 11 distinct nets for a cube. Some examples include:
- Four squares in a row with one square on top of the second and one below the third.
- Three squares in a row with one square on top of the middle and two attached to the sides.

Diagram: Sketch at least two different net arrangements with labels.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
Riya observed a cube and a cylinder in her geometry box. She wants to know: (a) How many faces, edges, and vertices does a cube have? (b) If the cylinder has no vertices, how is it different from a cube in terms of edges?
Answer:
Problem Interpretation

We studied solid shapes like cubes and cylinders. A cube is a 3D shape with equal sides, while a cylinder has circular bases.

Mathematical Modeling
  • Cube: 6 faces, 12 edges, 8 vertices
  • Cylinder: 2 faces (circular), 1 curved face, 2 edges (circular), no vertices
Solution

The cube has more edges and vertices than a cylinder, which has only curved edges and no vertices.

Question 2:
A pyramid has a square base with 4 triangular faces. (a) How many faces, edges, and vertices does it have? (b) Compare it with a triangular prism studied in our textbook.
Answer:
Problem Interpretation

Our textbook shows pyramids and prisms. A square pyramid has a base and triangular faces meeting at a point.

Mathematical Modeling
  • Square Pyramid: 5 faces (1 base + 4 triangles), 8 edges, 5 vertices
  • Triangular Prism: 5 faces (2 triangles + 3 rectangles), 9 edges, 6 vertices
Solution

The pyramid has fewer edges and vertices than the prism, which has rectangular faces and more edges.

Question 3:
Riya has a cube and a cylinder in her geometry box. She wants to find the number of faces, edges, and vertices for each shape. Help her by answering the following:
1. List the properties of a cube.
2. Compare them with a cylinder.
Answer:
Problem Interpretation

We studied that a cube is a 3D shape with equal sides, while a cylinder has circular bases.


Mathematical Modeling
  • Cube: Faces: 6, Edges: 12, Vertices: 8
  • Cylinder: Faces: 3 (2 circular, 1 curved), Edges: 2 (circular), Vertices: 0

Solution

Our textbook shows that cubes have flat faces, while cylinders have curved surfaces. This affects their edges and vertices.

Question 4:
A net of a cuboid is given with dimensions 4 cm × 3 cm × 2 cm. Identify:
1. The number of rectangles in the net.
2. The possible arrangement of these rectangles.
Answer:
Problem Interpretation

We learned that a cuboid net is a 2D layout of its 6 rectangular faces.


Mathematical Modeling
  • Rectangles: 6 (3 pairs of identical rectangles)
  • Arrangement: Can vary, but must include 4 cm × 3 cm, 4 cm × 2 cm, and 3 cm × 2 cm pairs.

Solution

Our textbook shows different net arrangements, but all must account for the cuboid's length, breadth, and height.

Question 5:
Riya observed a cube and a cylinder in her geometry box. She wondered how many faces, edges, and vertices each shape has. Help her by comparing these properties for both shapes.
Answer:
Problem Interpretation

We studied that a cube and cylinder are 3D shapes with different properties.

Mathematical Modeling
  • Cube: 6 faces, 12 edges, 8 vertices
  • Cylinder: 3 faces (2 circular, 1 curved), 2 edges, 0 vertices
Solution

Our textbook shows that a cube has more edges and vertices than a cylinder, which is curved and lacks vertices.

Question 6:
A tent is shaped like a triangular prism. If its base is an equilateral triangle with side 4 m and height 3 m, and the tent length is 6 m, find its total surface area.
Answer:
Problem Interpretation

We need to calculate the surface area of a triangular prism-shaped tent.

Mathematical Modeling
  • Base area = (√3/4) × side² = (√3/4) × 16 = 4√3 m²
  • Lateral area = Perimeter × height = (4×3) × 6 = 72 m²
Solution

Total surface area = 2 × base area + lateral area = 8√3 + 72 m². Our textbook shows similar prism examples.

Question 7:
Riya has a cube and a cylinder in her geometry box. She wants to find the number of faces, edges, and vertices for each shape. Help her by answering the following:
  • List the properties of a cube.
  • Compare them with a cylinder.
Answer:
Problem Interpretation

We studied that a cube is a 3D shape with equal sides. A cylinder has circular bases.


Mathematical Modeling
  • Cube: 6 faces (squares), 12 edges, 8 vertices.
  • Cylinder: 2 circular faces, 1 curved face, 2 edges (circular), no vertices.

Solution

Our textbook shows that cubes have flat faces, while cylinders have curved surfaces. This helps distinguish their properties.

Question 8:
A pyramid has a square base. Identify its:
  • Number of triangular faces.
  • Total edges and vertices.
Also, explain how it differs from a triangular prism.
Answer:
Problem Interpretation

We know a square-based pyramid has a base and triangular sides. A triangular prism has two triangular bases.


Mathematical Modeling
  • Pyramid: 4 triangular faces, 8 edges (4 base + 4 lateral), 5 vertices (1 apex + 4 base).
  • Triangular Prism: 2 triangular faces, 3 rectangular faces, 9 edges, 6 vertices.

Solution

Our textbook shows pyramids converge to an apex, while prisms have parallel bases. This changes their structure.

Question 9:
Riya has a cube and a cylinder in her geometry box. She wants to know how many faces, edges, and vertices each shape has. Help her by comparing these properties for both shapes.
Answer:
Problem Interpretation

We studied that a cube and a cylinder are 3D shapes with different properties.

Mathematical Modeling
  • Cube: 6 faces, 12 edges, 8 vertices
  • Cylinder: 3 faces (2 circular, 1 curved), 2 edges, 0 vertices
Solution

Our textbook shows that a cube has more edges and vertices than a cylinder, which is curved and lacks vertices.

Question 10:
A toy box is shaped like a cuboid with dimensions 10 cm × 5 cm × 3 cm. How many unit cubes of side 1 cm can fit inside it? Explain your steps.
Answer:
Problem Interpretation

We need to find how many 1 cm cubes fit in a cuboid of given dimensions.

Mathematical Modeling
  • Volume of toy box = 10 × 5 × 3 = 150 cm³
  • Volume of unit cube = 1 × 1 × 1 = 1 cm³
Solution

Our textbook shows that dividing the box's volume by the cube's volume gives the number of cubes: 150 ÷ 1 = 150 cubes.

Question 11:

Riya observed a solid shape in her geometry box. It has 6 faces, 12 edges, and 8 vertices. She wants to identify the shape and draw its 2D net.

a) Name the solid shape Riya observed.
b) Draw any one possible 2D net for this shape.

Answer:

a) The solid shape Riya observed is a cube because it has 6 faces, 12 edges, and 8 vertices, which are the defining properties of a cube.

b) One possible 2D net for a cube is shown below (diagrammatic representation):

Imagine unfolding the cube to form a cross-like shape with 6 squares arranged as follows:

  • One square in the center.
  • Four squares attached to the top, bottom, left, and right sides of the central square.
  • One square attached to the right side of the top square.

This net can be folded back into a cube by joining the edges appropriately.

Question 12:

During a school activity, students were given clay to model different solid shapes. Aarav made a shape with 1 curved face, 1 flat circular face, and no edges or vertices.

a) Identify the solid shape Aarav created.
b) Give one real-life example of this shape.

Answer:

a) Aarav created a cone because it has 1 curved face, 1 flat circular face, and no edges or vertices.

b) A real-life example of a cone is an ice-cream cone or a birthday party hat. Both have a circular base and a curved surface tapering to a point.

Question 13:
Riya observed a solid shape in her geometry box. It has 6 faces, 12 edges, and 8 vertices.

Identify the shape and draw its net.

Also, explain how the net helps in visualising the 3D shape.

Answer:

The given solid is a cube because it has 6 square faces, 12 edges, and 8 vertices, which are properties of a cube.

Here’s how to draw its net:

  • Draw 6 squares arranged in a cross-like pattern (T-shape or other valid net arrangement).

A net helps in visualising a 3D shape by:

  • Showing all faces of the shape in a 2D layout.
  • Demonstrating how folding along edges forms the 3D object.
  • Making it easier to calculate surface area.

For example, the net of a cube clearly shows how the squares connect to form the closed box shape.

Question 14:
A cylindrical container has a height of 10 cm and a base radius of 3.5 cm.

Draw its 2D net and calculate the total surface area using the net.

Explain why the net of a cylinder includes a rectangle and two circles.

Answer:

The net of a cylinder consists of:

  • One rectangle (representing the curved surface).
  • Two circles (representing the top and bottom bases).

Steps to calculate total surface area:


1. Area of the rectangle (curved surface) = height × circumference = 10 cm × (2 × π × 3.5 cm) = 220 cm² (using π = 22/7).
2. Area of two circles = 2 × π × radius² = 2 × (22/7) × 3.5 × 3.5 = 77 cm².
3. Total surface area = 220 cm² + 77 cm² = 297 cm².

The net includes a rectangle because when unrolled, the curved surface becomes a rectangle. The two circles represent the flat circular bases.

Question 15:
Riya has a set of solid shapes in her geometry box: a cube, a sphere, and a cylinder. She wants to identify the number of faces, edges, and vertices for each shape. Help her by providing the details for all three shapes.
Answer:

Here are the details of the faces, edges, and vertices for each shape:

  • Cube:
    Faces: 6 (all square-shaped)
    Edges: 12 (where two faces meet)
    Vertices: 8 (corners where edges meet)
  • Sphere:
    Faces: 1 (curved surface)
    Edges: 0 (no straight edges)
    Vertices: 0 (no corners)
  • Cylinder:
    Faces: 3 (2 circular bases and 1 curved surface)
    Edges: 2 (circular edges where the bases meet the curved surface)
    Vertices: 0 (no corners)

Remember, a face is a flat surface, an edge is where two faces meet, and a vertex is a corner point.

Question 16:
A net of a 3D shape is given below with squares and triangles. When folded, it forms a square pyramid. Identify the number of squares and triangles in the net and explain how they form the pyramid.
Answer:

A square pyramid is formed using the following net components:

  • 1 square (the base of the pyramid)
  • 4 triangles (the lateral faces that meet at the apex)

When folded:
The square forms the flat base of the pyramid.
The four triangles fold upwards and meet at a single point (the apex), creating the pyramid's sloping sides.
This structure has 5 faces in total (1 square + 4 triangles), 8 edges, and 5 vertices.

Visualization tip: Imagine the square lying flat, and the triangles rising to form a pointed top, like a pyramid.

Question 17:
Riya has a set of 3D shapes: a cube, a cylinder, and a cone. She wants to identify the number of faces, edges, and vertices for each shape. Help her by filling the details in a tabular form and explain why a cylinder has no vertex.
Answer:

Here is the table comparing the properties of the given 3D shapes:


ShapeFacesEdgesVertices
Cube6128
Cylinder3 (2 circular, 1 curved)2 (circular edges)0
Cone2 (1 circular base, 1 curved)1 (circular edge)1 (apex)

A cylinder has no vertex because its surfaces are curved and circular, meeting smoothly without any sharp corners or points. Unlike a cube or cone, which have distinct edges and corners, a cylinder's edges are circular and continuous, eliminating the need for vertices.

Question 18:
A net of a 3D shape is given below with squares and triangles. Identify the shape formed when this net is folded and explain how you determined it. Also, state one real-life example of this shape.
Answer:

The given net consists of 1 square base and 4 triangular faces attached to its sides. When folded, it forms a square pyramid.


Steps to identify:
1. The square acts as the base.
2. The four triangles meet at a common point (apex) above the base.
3. This structure matches the definition of a square pyramid.


Real-life example: The Great Pyramid of Giza or a pyramid-shaped tent.

Question 19:
Riya has a set of 3D shapes including a cube, a cylinder, and a cone. She wants to identify the number of faces, edges, and vertices for each shape. Help her by providing the details for all three shapes.
Answer:

Here are the details of the faces, edges, and vertices for each shape:

  • Cube:
    Faces: 6 (all square-shaped)
    Edges: 12
    Vertices: 8
  • Cylinder:
    Faces: 3 (2 circular bases and 1 curved surface)
    Edges: 2 (circular edges of the bases)
    Vertices: 0 (since it has curved surfaces)
  • Cone:
    Faces: 2 (1 circular base and 1 curved surface)
    Edges: 1 (circular edge of the base)
    Vertices: 1 (the apex point)

Remember, a vertex is a corner point where edges meet, and an edge is a line segment where two faces meet.

Question 20:
A net of a 3D shape is given below with 6 squares arranged in a cross pattern. Identify the shape formed when this net is folded and also write one real-life example of this shape.
Answer:

The given net with 6 squares arranged in a cross pattern forms a cube when folded.


Steps to visualize the folding:
1. The central square acts as the base.
2. The four adjacent squares fold upwards to form the sides.
3. The top square folds to cover the cube.


A real-life example of a cube is a Rubik's Cube or a dice used in board games.


Note: A cube has all sides equal, and its net always consists of 6 squares connected in a way that allows folding into a closed 3D shape.

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