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Boxes and Sketches – CBSE NCERT Study Resources

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5th

5th - Mathematics (Math-Magic)

Boxes and Sketches

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Overview of the Chapter

This chapter, "Boxes and Sketches," introduces students to the concepts of visualizing and drawing 3D shapes on a 2D plane. It helps them understand how objects appear from different perspectives and how to represent them through sketches. The chapter also covers the basics of nets, which are 2D layouts that can be folded to form 3D shapes.

Key Concepts

3D Shapes and Their Views

Students learn about different views of 3D objects, such as the top view, front view, and side view. They practice sketching these views for simple objects like cubes, cuboids, and cylinders.

Nets of 3D Shapes

A net is a 2D shape that can be folded to form a 3D object. Students explore nets for cubes, cuboids, and other simple shapes, understanding how folding creates the 3D structure.

Drawing Boxes

Students practice drawing boxes (cuboids) from different angles, learning how to represent depth and perspective in their sketches.

Activities and Exercises

  • Identifying and drawing the top, front, and side views of given objects.
  • Creating nets for cubes and cuboids and folding them to form the 3D shapes.
  • Drawing boxes in different orientations to understand perspective.

Learning Outcomes

By the end of this chapter, students will be able to:

  • Visualize and sketch 3D objects from different views.
  • Understand and create nets for simple 3D shapes.
  • Draw boxes and other simple shapes with perspective.

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:
What is the shape of the net of a cube?
Answer:

The net of a cube is a 2D shape made of 6 squares arranged in a cross or other patterns that can fold into a cube.

Question 2:
How many edges does a cuboid have?
Answer:

A cuboid has 12 edges.

Question 3:
Name the 3D shape formed by folding the net with two circles and one rectangle.
Answer:

Folding this net forms a cylinder, where the circles are the bases and the rectangle wraps around.

Question 4:
What is the difference between a 2D shape and a 3D shape?
Answer:

2D shapes are flat with only length and width (e.g., square), while 3D shapes have length, width, and height (e.g., cube).

Question 5:
Draw the net of a square pyramid.
Answer:

The net consists of:
1 square (base)
4 triangles (lateral faces)
Note: Diagrams are essential for full marks.

Question 6:
How many faces does a triangular prism have?
Answer:

A triangular prism has 5 faces: 2 triangular and 3 rectangular.

Question 7:
Identify the shape whose net has 4 equilateral triangles.
Answer:

This net forms a tetrahedron, a pyramid with a triangular base.

Question 8:
What is the net of a cone?
Answer:

The net of a cone has:
1 circle (base)
1 sector of a circle (lateral surface)

Question 9:
Can a cylinder roll? Why or why not?
Answer:

Yes, a cylinder can roll because its curved surface allows smooth movement.

Question 10:
Name a real-life object shaped like a cuboid.
Answer:

A book or brick is shaped like a cuboid.

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:
What is the difference between a 2D shape and a 3D shape? Give one example of each.
Answer:

A 2D shape has only length and width (like a rectangle), while a 3D shape has length, width, and height (like a cube).


Example of 2D: Square
Example of 3D: Cuboid
Question 2:
How many faces does a cube have?
Answer:

A cube has 6 faces, all of which are squares of equal size.

Question 3:
What is the shape of the base of a cylinder?
Answer:

The base of a cylinder is a circle. It has two circular bases, one at the top and one at the bottom.

Question 4:
Name the 3D shape that has only one curved surface and no edges or vertices.
Answer:

A sphere has only one curved surface and no edges or vertices. Examples include a football or a globe.

Question 5:
Draw the top view of a rectangular prism.
Answer:

The top view of a rectangular prism is a rectangle.
[Diagram: A simple rectangle labeled as 'Top View of Rectangular Prism']

Question 6:
How many edges does a triangular pyramid have?
Answer:

A triangular pyramid (or tetrahedron) has 6 edges.

  • 3 edges form the base triangle.
  • 3 edges connect the base to the apex.

Question 7:
How many edges does a cube have?
Answer:

A cube has 12 edges.


Edges are the straight lines where two faces meet.

Question 8:
Name the 3D shape that has 6 rectangular faces.
Answer:

The 3D shape with 6 rectangular faces is a cuboid.


Example: A shoebox.

Question 9:
What is the net of a 3D shape? Draw a simple net for a cube.
Answer:

An net is a 2D layout that can be folded to form a 3D shape.


Net of a cube:


Draw 6 squares arranged like a cross (one square in the center with four attached to its sides and one above or below).

Question 10:
Identify the 3D shape formed by stacking circles of the same size one above the other.
Answer:

Stacking circles forms a cylinder.


Example: A soda can.

Question 11:
How many vertices does a triangular pyramid have?
Answer:

A triangular pyramid has 4 vertices.


Vertices are the corner points where edges meet.

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 net that can form a cube. Draw and explain why the other options cannot form a cube.
Answer:

A net is a 2D shape that can be folded to form a 3D object like a cube. The correct net for a cube has 6 squares arranged in a way that all faces meet without overlaps.


For example, a 'T-shaped' net with 6 squares can form a cube.


Other nets may fail because:

  • Some have squares overlapping when folded.
  • Others have gaps or missing faces.
  • Incorrect arrangements may leave some squares unattached.
Question 2:
Explain how to sketch the 2D top view of a rectangular box placed on a table. What will you see from above?
Answer:

To sketch the 2D top view of a rectangular box, imagine looking directly down at it from above.


You will see:

  • A rectangle representing the top face of the box.
  • No sides or depth, just the flat surface.

If the box has a lid or design, those details will also be visible in the top view.

Question 3:
A box has dimensions 5 cm × 3 cm × 2 cm. How many unit cubes of 1 cm³ can fit inside it? Show the steps.
Answer:

To find the number of unit cubes (1 cm³ each) that fit in the box:


Step 1: Multiply the length, width, and height.
Step 2: 5 cm × 3 cm × 2 cm = 30 cm³.
Step 3: Since each cube is 1 cm³, 30 cubes can fit.


The box can hold 30 unit cubes.

Question 4:
Differentiate between a 2D shape and a 3D shape using the example of a square and a cube.
Answer:

2D shapes like a square have only length and width. They are flat and cannot be held.


3D shapes like a cube have length, width, and height. They are solid and can be held.


Key differences:

  • A square has 4 sides; a cube has 6 faces.
  • A square has area; a cube has volume.
Question 5:
Draw the front view, side view, and top view of a cylindrical box placed upright. Label each view.
Answer:

For a cylindrical box:


Front view: A rectangle (height × diameter).
Side view: Same as front view (another rectangle).
Top view: A circle (the circular face).


Labels:
1. Front/Side: Rectangle labeled 'Height × Diameter'.
2. Top: Circle labeled 'Top View'.

Question 6:
What is a net of a 3D shape? Give an example of a net for a cube.
Answer:

A net is a 2D shape that can be folded to form a 3D object. It shows all the faces of the 3D shape laid out flat.


Example of a net for a cube:


  • A cube has 6 square faces.
  • One common net is a cross-shaped arrangement of 6 squares.
  • When folded, the squares form the sides, top, bottom, front, and back of the cube.
Question 7:
How many edges does a cuboid have? Explain with a diagram.
Answer:

A cuboid has 12 edges.


Explanation:


  • A cuboid has 6 faces (all rectangles).
  • Each face has 4 edges, but edges are shared between faces.
  • Total edges = 4 (top) + 4 (bottom) + 4 (vertical) = 12.

Diagram: Imagine a rectangular box with edges along its length, width, and height.

Question 8:
Draw the top, front, and side views of a cylinder.
Answer:

Views of a cylinder:


  • Top view: A circle (since the top face is circular).
  • Front view: A rectangle (showing the height and curved surface).
  • Side view: Same as the front view—a rectangle.

Note: The rectangle's width in front/side views represents the diameter of the cylinder.

Question 9:
What is the difference between a cube and a cuboid? Give one real-life example of each.
Answer:

Difference between a cube and a cuboid:


  • A cube has all faces as squares and all edges equal.
  • A cuboid has rectangular faces and edges of different lengths.

Examples:


  • Cube: A Rubik's cube or dice.
  • Cuboid: A book or a brick.
Question 10:
Explain how to identify the base, faces, and edges of a pyramid with a square base.
Answer:

For a pyramid with a square base:


  • Base: The square at the bottom (1 face).
  • Faces: 4 triangular faces (sides) + 1 square base = 5 faces total.
  • Edges: 4 edges of the base + 4 edges connecting the base to the apex (top point) = 8 edges.

Tip: The apex is where all triangular faces meet.

Question 11:
Identify the net that can form a cube. Draw and explain why the chosen net is correct.
Answer:

A net is a 2D shape that can be folded to form a 3D object like a cube. The correct net for a cube has 6 squares arranged in a way that all faces meet without overlapping.


Example of a correct net:


  • Four squares in a row.
  • One square attached above the second square.
  • One square attached below the third square.

When folded, this forms a cube because all six faces are connected properly.

Question 12:
Explain how to sketch the 2D top view of a rectangular box placed on a table.
Answer:

The 2D top view of a rectangular box shows how it looks from directly above. Here’s how to sketch it:


1. Draw a rectangle to represent the top face of the box.
2. Ensure the sides are proportional to the actual box.
3. Do not include the side or front faces in this view.


This view helps visualize the box’s dimensions from above.

Question 13:
What is the difference between a 2D shape and a 3D shape? Give an example of each.
Answer:

2D shapes are flat and have only length and width, like a square or circle.
3D shapes have length, width, and height, like a cube or sphere.


Example:
- 2D: A triangle drawn on paper.
- 3D: A pyramid you can hold.


3D shapes occupy space, while 2D shapes do not.

Question 14:
Draw the front view of a cylindrical box placed upright. Explain your drawing.
Answer:

The front view of a cylindrical box looks like a rectangle because:


1. The circular top and bottom faces appear as straight lines from the front.
2. The curved side appears as two vertical lines.
3. The height of the rectangle matches the box’s height.


This view simplifies the 3D shape into a 2D representation.

Question 15:
How many faces, edges, and vertices does a rectangular prism have? Show the steps to calculate each.
Answer:

A rectangular prism has:


1. Faces: 6 (top, bottom, front, back, left, right).
2. Edges: 12 (4 edges on top, 4 on bottom, and 4 vertical).
3. Vertices: 8 (corners where edges meet).


Steps:
- Count each flat surface for faces.
- Count where two faces meet for edges.
- Count where edges meet for vertices.

Question 16:
How many faces, edges, and vertices does a cuboid have?
Answer:

A cuboid has:

  • 6 faces (all rectangular)
  • 12 edges (where two faces meet)
  • 8 vertices (corners where edges meet)

Question 17:
Explain how to draw the net of a cube with the help of a diagram.
Answer:

To draw a net of a cube:
1. Draw 6 squares connected edge-to-edge in a way that they can be folded to form a cube.
2. Common net arrangements include a T-shape or a cross.
Diagram: Imagine 4 squares in a row, with one square attached above the second square and one below the third square.

Question 18:
Why is it important to learn about boxes and sketches in daily life?
Answer:

Learning about boxes and sketches helps us:

  • Understand packaging designs (like cereal boxes)
  • Visualize buildings or objects before making them
  • Improve spatial thinking for maps or art
It makes real-world problem-solving easier!

Question 19:
If a box has a length of 5 cm, width of 3 cm, and height of 2 cm, what is its total surface area?
Answer:

Total surface area = 2 × (length×width + width×height + height×length)
= 2 × (5×3 + 3×2 + 2×5)
= 2 × (15 + 6 + 10)
= 2 × 31
= 62 cm²

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 how to draw the 2D net of a cube and describe the steps to fold it into a 3D shape. Also, mention how many different nets are possible for a cube.
Answer:

A 2D net of a cube is a flat shape that can be folded to form the cube. Here are the steps to draw and fold it:


Step 1: Drawing the Net
Draw six squares arranged in a way that they can be folded to form a cube. One common net looks like a cross with one square in the center and four squares attached to its sides, and the sixth square attached to one of them.

Step 2: Folding the Net
1. Cut out the net carefully along the edges.
2. Fold along the edges where the squares meet.
3. Bring the squares together to form the cube's sides.
4. Use glue or tape to secure the edges.

There are 11 different nets possible for a cube, each with a unique arrangement of squares. These nets help us visualize how a 3D shape can be formed from a 2D layout.

Question 2:
A box has dimensions of 5 cm × 3 cm × 2 cm. Draw its 2D net and calculate the total surface area. Explain your steps clearly.
Answer:

To draw the 2D net of the box and calculate its surface area, follow these steps:


Step 1: Drawing the Net
The box has three pairs of rectangular faces:
1. Two faces of 5 cm × 3 cm (top and bottom).
2. Two faces of 5 cm × 2 cm (front and back).
3. Two faces of 3 cm × 2 cm (left and right sides).
Arrange these rectangles in a net so they can be folded into the box.

Step 2: Calculating Surface Area
1. Area of one 5 cm × 3 cm face = 5 × 3 = 15 cm².
Total for two such faces = 15 × 2 = 30 cm².
2. Area of one 5 cm × 2 cm face = 5 × 2 = 10 cm².
Total for two such faces = 10 × 2 = 20 cm².
3. Area of one 3 cm × 2 cm face = 3 × 2 = 6 cm².
Total for two such faces = 6 × 2 = 12 cm².

Total Surface Area = 30 + 20 + 12 = 62 cm².
Question 3:
Rahul has a rectangular box with dimensions 10 cm × 6 cm × 4 cm. He wants to create a 2D sketch (net) of this box to visualize how it can be folded. Help him by drawing the correct net and explaining the steps involved.
Answer:

To draw the net of Rahul's rectangular box, follow these steps:


Step 1: Understand the dimensions
The box has dimensions 10 cm (length) × 6 cm (width) × 4 cm (height).

Step 2: Plan the net layout
A rectangular box has 6 faces. The net will consist of these faces arranged in a way that they can be folded to form the box.

Step 3: Draw the net
Here’s one possible net arrangement:
  • Draw a central rectangle for the base (10 cm × 6 cm).
  • Attach the front (10 cm × 4 cm) and back (10 cm × 4 cm) rectangles above and below the base.
  • On either side of the base, attach the left (6 cm × 4 cm) and right (6 cm × 4 cm) rectangles.
  • Finally, attach the top (10 cm × 6 cm) rectangle to one of the side faces.

Step 4: Verify the net
Ensure all faces are correctly sized and connected so folding them forms the box. Labeling dimensions helps avoid confusion.

Remember, nets can vary in arrangement, but all must include the 6 faces of the box.

Question 4:
Priya has a cube-shaped box with each edge measuring 5 cm. She wants to wrap it with gift paper. Calculate the total area of the gift paper needed and explain how you arrived at the answer.
Answer:

To find the total area of gift paper needed to wrap Priya's cube-shaped box, follow these steps:


Step 1: Identify the shape and dimensions
The box is a cube with each edge = 5 cm.

Step 2: Recall the formula
A cube has 6 identical square faces. The total surface area = 6 × (side)2.

Step 3: Calculate the area of one face
Area of one square face = side × side = 5 cm × 5 cm = 25 cm2.

Step 4: Calculate the total surface area
Total surface area = 6 × 25 cm2 = 150 cm2.

Thus, Priya needs 150 cm2 of gift paper to wrap the entire box. This ensures all 6 faces are covered.


Note: If the box has overlapping flaps or extra paper for folding, the required area may slightly increase.

Question 5:
Explain how to draw the 2D sketch of a cube and label its parts. Also, describe the difference between a net of a cube and its 2D sketch.
Answer:

To draw a 2D sketch of a cube, follow these steps:
1. Draw a square to represent the front face of the cube.
2. Draw another square of the same size adjacent to the first one, slightly tilted, to represent the side face.
3. Connect the corresponding corners of both squares with straight lines to show depth.
4. Label the faces as front, side, and top for clarity.

The difference between a net of a cube and its 2D sketch is:
- A net is a flattened 2D layout of all 6 faces of the cube, which can be folded to form the 3D shape.
- A 2D sketch shows only 3 visible faces of the cube in a perspective view, giving an illusion of depth.

For example, a net has 6 squares arranged in a cross or other pattern, while a 2D sketch shows overlapping squares to represent the cube's edges and corners.

Question 6:
A box has dimensions 5 cm (length) × 3 cm (width) × 2 cm (height). Draw its net and calculate the total surface area. Explain each step clearly.
Answer:

To draw the net of the box:
1. Since the box is a rectangular prism, its net consists of 6 rectangles.
2. Arrange them in a way that opposite faces are of the same size:
- Two rectangles of 5 cm × 3 cm (top and bottom faces).
- Two rectangles of 5 cm × 2 cm (front and back faces).
- Two rectangles of 3 cm × 2 cm (side faces).
3. Draw these rectangles connected edge-to-edge, forming a cross or another valid net pattern.

To calculate the total surface area:
1. Find the area of each pair of opposite faces:
- Top and bottom: 2 × (5 cm × 3 cm) = 30 cm².
- Front and back: 2 × (5 cm × 2 cm) = 20 cm².
- Sides: 2 × (3 cm × 2 cm) = 12 cm².
2. Add all the areas: 30 cm² + 20 cm² + 12 cm² = 62 cm².

Thus, the total surface area of the box is 62 cm².

Question 7:
Explain how to draw the 2D net of a cube and describe its importance in understanding 3D shapes. Provide step-by-step instructions with a diagram.
Answer:

A net of a cube is a 2D shape that can be folded to form the 3D cube. It helps us visualize how flat surfaces come together to form solid objects.


Steps to draw the net of a cube:


1. Draw a square in the center. This will be the base of the cube.
2. Draw four equal squares attached to each side of the central square, forming a '+' shape.
3. Draw one more square attached to any one of the outer squares. This will be the lid.

Diagram:


[Imagine a cross-shaped net with 6 squares: one in the center, four on each side, and one attached to the top square.]

Importance:

  • Helps understand how 2D shapes form 3D objects.
  • Makes it easier to calculate surface area.
  • Used in packaging and design.
Question 8:
A box has dimensions 5 cm (length) × 3 cm (width) × 2 cm (height). Draw its possible nets and explain how you determined them.
Answer:

A net is a 2D layout of a 3D box that can be folded to form the box. For a box with dimensions 5 cm × 3 cm × 2 cm, there can be multiple net variations.


Steps to determine possible nets:


1. Identify the faces: The box has 6 faces (2 of each pair: 5×3, 5×2, 3×2).
2. Arrange the rectangles in a way that they can fold into a box without overlaps.

Example of one net:


[Imagine a layout where:
- A 5×3 rectangle is in the center.
- A 5×2 rectangle is attached to its top and bottom.
- A 3×2 rectangle is attached to its left and right.]

Key points:

  • Nets must have all 6 faces connected edge-to-edge.
  • They must fold without gaps or overlaps.
  • Different arrangements are possible, but all must follow the box's dimensions.
Question 9:
Rahul has a rectangular box with dimensions 10 cm × 6 cm × 4 cm. He wants to create a 2D sketch (net) of this box to visualize how it can be folded. Help him by drawing the correct net and explaining the steps involved in creating it.
Answer:

To draw the net of Rahul's rectangular box, follow these steps:


Step 1: Understand the dimensions
The box has dimensions 10 cm (length) × 6 cm (width) × 4 cm (height).

Step 2: Identify the faces
A rectangular box has 6 faces: front, back, left, right, top, and bottom.

Step 3: Draw the net layout
Here’s one possible net arrangement:
  • Draw a central rectangle for the front face (10 cm × 4 cm).
  • Attach the right face (6 cm × 4 cm) to its right.
  • Place the back face (10 cm × 4 cm) to the right of the right face.
  • Draw the left face (6 cm × 4 cm) to the left of the front face.
  • Above the front face, draw the top face (10 cm × 6 cm).
  • Below the front face, draw the bottom face (10 cm × 6 cm).

Step 4: Label the dimensions
Ensure all sides are labeled correctly to avoid confusion while folding.

Step 5: Verify the net
Check that the total dimensions add up correctly and all faces are connected properly for folding.
Question 10:
Priya has a cube-shaped box with each edge measuring 5 cm. She wants to wrap it as a gift. Explain how she can calculate the total surface area of the box and determine the minimum amount of wrapping paper needed.
Answer:

To calculate the total surface area of Priya's cube-shaped box, follow these steps:


Step 1: Understand the shape
A cube has 6 identical square faces.

Step 2: Find the area of one face
Area of one square face = side × side = 5 cm × 5 cm = 25 cm².

Step 3: Calculate total surface area
Total surface area = 6 × area of one face = 6 × 25 cm² = 150 cm².

Step 4: Determine wrapping paper needed
Since the minimum wrapping paper required must cover all faces, Priya needs at least 150 cm² of paper.

Additional Tip: If the wrapping paper has patterns or needs extra for overlap, Priya should account for that too. For example, adding 10% extra would mean 150 cm² + 15 cm² = 165 cm².

Question 11:
Explain how to draw the 2D sketch of a cube using the oblique sketching method. Include the steps and mention any important observations.
Answer:

To draw a 2D sketch of a cube using the oblique sketching method, follow these steps:


  • Step 1: Draw a square to represent the front face of the cube. Ensure all sides are equal and angles are 90°.

  • Step 2: From the top-right corner of the square, draw a line at a 45° angle to the right. This line represents the depth of the cube.

  • Step 3: From the bottom-right corner of the square, draw another line at the same 45° angle and of the same length as the previous line.

  • Step 4: Connect the ends of these two lines with a parallel line to complete the back face of the cube.

  • Step 5: Draw the top and side faces by connecting the corresponding corners of the front and back faces.

Observation: In oblique sketching, the front face remains unchanged, while the depth lines are drawn at an angle (usually 45°). This method gives a 3D-like appearance to the 2D sketch.

Question 12:
A box has dimensions 5 cm × 3 cm × 2 cm. Draw its net and label all the sides. Explain how the net helps in understanding the 3D shape.
Answer:

The net of a box with dimensions 5 cm × 3 cm × 2 cm can be drawn as follows:


  • Step 1: Draw a rectangle for the base with dimensions 5 cm (length) × 3 cm (width). Label it as the bottom face.

  • Step 2: Attach four rectangles around the base: two with dimensions 5 cm × 2 cm (for the front and back faces) and two with dimensions 3 cm × 2 cm (for the left and right faces).

  • Step 3: Draw another 5 cm × 3 cm rectangle attached to one of the 5 cm × 2 cm rectangles. Label it as the top face.

Explanation: A net is a 2D layout of all the faces of a 3D shape. When folded along the edges, it forms the actual 3D box. Nets help visualize how the faces are connected and understand the surface area of the shape. For example, the total surface area of this box can be calculated by adding the areas of all six rectangles in the net.

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 has a cubical box with each edge measuring 5 cm. She wants to cover it with colored paper. Calculate the total area of the paper needed to cover the box completely. Also, explain why the formula for the total surface area of a cube is 6 × (edge)².
Answer:

To find the total area of the paper needed to cover the cubical box, we use the formula for the total surface area of a cube: 6 × (edge)².

Given: Edge of the cube = 5 cm.

Step 1: Calculate the area of one face of the cube.
Area of one face = (edge)² = 5 cm × 5 cm = 25 cm².

Step 2: Multiply by 6 since a cube has 6 identical faces.
Total surface area = 6 × 25 cm² = 150 cm².

Explanation of the formula:
A cube has 6 identical square faces. The area of one face is (edge)². To cover all faces, we multiply by 6. Hence, the formula is 6 × (edge)².

Question 2:
A rectangular prism has dimensions 8 cm (length), 6 cm (width), and 4 cm (height). Draw its net and calculate the total surface area. Explain how the net helps visualize the 3D shape.
Answer:

To calculate the total surface area of the rectangular prism, we use the formula: 2 × (lw + lh + wh).

Given: Length (l) = 8 cm, Width (w) = 6 cm, Height (h) = 4 cm.

Step 1: Calculate the area of the three unique faces.
Area of the first face (l × w) = 8 cm × 6 cm = 48 cm².
Area of the second face (l × h) = 8 cm × 4 cm = 32 cm².
Area of the third face (w × h) = 6 cm × 4 cm = 24 cm².

Step 2: Add these areas and multiply by 2.
Total surface area = 2 × (48 + 32 + 24) = 2 × 104 = 208 cm².

Explanation of the net:
A net is a 2D layout of all the faces of a 3D shape. For the rectangular prism, the net consists of 6 rectangles (3 pairs of identical rectangles). It helps us visualize how the 3D shape can be folded from a flat surface.

Question 3:
A box is in the shape of a cylinder with a radius of 3 cm and a height of 10 cm. Calculate the curved surface area and the total surface area (including the top and bottom). Use π = 3.14.
Answer:

To find the curved surface area and total surface area of the cylindrical box, we use the formulas:

Curved surface area = 2πrh.
Total surface area = 2πr(r + h).

Given: Radius (r) = 3 cm, Height (h) = 10 cm, π = 3.14.

Step 1: Calculate the curved surface area.
Curved surface area = 2 × 3.14 × 3 cm × 10 cm = 188.4 cm².

Step 2: Calculate the area of the top and bottom circles.
Area of one circle = πr² = 3.14 × 3 cm × 3 cm = 28.26 cm².
Area of two circles = 2 × 28.26 cm² = 56.52 cm².

Step 3: Add to get the total surface area.
Total surface area = Curved surface area + Area of two circles = 188.4 cm² + 56.52 cm² = 244.92 cm².

The curved surface area covers the side of the cylinder, while the total surface area includes the top and bottom circular faces.

Question 4:
Riya has a cubical box with each edge measuring 5 cm. She wants to cover it with colored paper.

Calculate the total area of paper needed to cover the box completely.

Also, explain why knowing the surface area is important in real-life situations.
Answer:

A cubical box has 6 equal square faces. To find the total area of paper needed, we calculate the surface area of the cube.


Step 1: Area of one face = side × side = 5 cm × 5 cm = 25 cm².
Step 2: Total surface area = 6 × area of one face = 6 × 25 cm² = 150 cm².

Riya needs 150 cm² of colored paper.

Knowing the surface area is important because:

  • It helps in estimating materials needed for wrapping or painting objects.
  • It is used in designing packaging to minimize waste.
  • It helps in understanding heat or cooling requirements in science.
Question 5:
A rectangular box has dimensions 8 cm (length), 6 cm (width), and 4 cm (height).

Draw a net of this box and label the dimensions. Also, calculate the total length of tape required to cover all its edges.

Answer:

A net of the rectangular box can be drawn as follows:


1. Draw a central rectangle for the base (8 cm × 6 cm).
2. Attach four rectangles around it for the sides: two of 8 cm × 4 cm (length × height) and two of 6 cm × 4 cm (width × height).
3. Label all dimensions clearly.

To calculate the total length of tape for all edges:


Step 1: A rectangular box has 12 edges: 4 edges of length, 4 of width, and 4 of height.
Step 2: Total tape length = (4 × 8 cm) + (4 × 6 cm) + (4 × 4 cm) = 32 cm + 24 cm + 16 cm = 72 cm.

72 cm of tape is required.

Question 6:
A box is in the shape of a triangular prism with a triangular base of sides 3 cm, 4 cm, and 5 cm, and a height of 10 cm.

Calculate the total surface area of the box. Also, identify the type of triangle formed by the base and justify your answer.

Answer:

To calculate the surface area of the triangular prism:


Step 1: Find the area of the triangular base. The sides 3 cm, 4 cm, and 5 cm form a right-angled triangle (since 3² + 4² = 5²).
Area of triangle = ½ × base × height = ½ × 3 cm × 4 cm = 6 cm².
Step 2: Perimeter of the base = 3 cm + 4 cm + 5 cm = 12 cm.
Step 3: Lateral surface area = perimeter × height = 12 cm × 10 cm = 120 cm².
Step 4: Total surface area = 2 × base area + lateral area = (2 × 6 cm²) + 120 cm² = 132 cm².

The total surface area is 132 cm².

The base is a right-angled triangle because it satisfies the Pythagorean theorem (3² + 4² = 5²).

Question 7:
Riya has a cubical box with each edge measuring 5 cm. She wants to cover it with colored paper.

a) How much paper (in cm²) is needed to cover the box completely?

b) If she cuts the box along one edge and unfolds it, what shape will the net of the box look like? Draw the net.

Answer:

a) To find the paper needed, calculate the total surface area of the cube.
Formula: Surface Area = 6 × (edge)²
= 6 × (5 cm)²
= 6 × 25 cm²
= 150 cm² of paper is required.

b) When unfolded, the net of a cube consists of 6 squares joined edge-to-edge.
Here’s how to draw it:
1. Draw a central square.
2. Attach 4 squares to its sides (top, bottom, left, right).
3. Add the 6th square to any of the attached squares (like a "T" shape).
Note: There are 11 possible net designs for a cube, but this is the simplest one!

Question 8:
A rectangular prism box has dimensions 8 cm (length), 6 cm (width), and 4 cm (height).

a) Calculate its volume.

b) How many smaller cubes of side 2 cm can fit inside it?

Answer:

a) Volume of the box = length × width × height
= 8 cm × 6 cm × 4 cm
= 192 cm³.

b) To fit smaller cubes:
1. Divide each dimension by the smaller cube’s side (2 cm):
- Lengthwise: 8 cm ÷ 2 cm = 4 cubes
- Widthwise: 6 cm ÷ 2 cm = 3 cubes
- Heightwise: 4 cm ÷ 2 cm = 2 cubes
2. Total cubes = 4 × 3 × 2 = 24 cubes can fit inside.

Question 9:
A box is designed with a triangular prism shape (base: equilateral triangle of side 3 cm, height: 10 cm).

a) Find the lateral surface area (excluding the triangular bases).

b) How many such boxes can fit into a larger cubical box of side 30 cm? Assume no gaps.

Answer:

a) Lateral Surface Area = Perimeter of base × height
1. Perimeter of equilateral triangle = 3 × side = 3 × 3 cm = 9 cm
2. Lateral Area = 9 cm × 10 cm = 90 cm².

b) To fit into the larger cube:
1. Volume of triangular prism box = Base Area × height
Base Area = (√3/4) × (side)² ≈ 3.9 cm²
Volume ≈ 3.9 cm² × 10 cm = 39 cm³
2. Volume of larger cube = 30 cm × 30 cm × 30 cm = 27,000 cm³
3. Number of boxes ≈ 27,000 cm³ ÷ 39 cm³ ≈ 692 boxes (rounded down).
Note: Actual packing may vary due to shape constraints.

Question 10:
Riya has a cubical box with each edge measuring 5 cm. She wants to cover it with colored paper. Calculate the total area of the paper needed to cover the entire box. Also, explain why knowing the surface area is important in real-life situations.
Answer:

To find the total area of the paper needed, we calculate the surface area of the cubical box.


Step 1: A cube has 6 identical square faces.
Step 2: Area of one face = side × side = 5 cm × 5 cm = 25 cm².
Step 3: Total surface area = 6 × area of one face = 6 × 25 cm² = 150 cm².

Knowing the surface area helps in real-life situations like:

  • Determining how much wrapping paper is needed for gifts.
  • Calculating the amount of paint required to cover an object.
  • Designing packaging materials efficiently.
Question 11:
A rectangular box has dimensions 8 cm (length), 6 cm (width), and 4 cm (height). Draw a net of this box and label the dimensions. Explain how the net helps in understanding the 3D shape.
Answer:

Here’s how to draw the net of the rectangular box:


Step 1: Draw a rectangle for the base (8 cm × 6 cm).
Step 2: Attach four rectangles around it for the sides:
  • Two rectangles of 8 cm × 4 cm (length × height).
  • Two rectangles of 6 cm × 4 cm (width × height).
Step 3: Add the top rectangle (8 cm × 6 cm) to complete the net.

The net helps in understanding the 3D shape by:

  • Showing all faces of the box in a 2D layout.
  • Making it easier to visualize how the box folds into its 3D form.
  • Helping calculate the surface area by seeing all sides at once.
Question 12:
A gift box is in the shape of a cylinder with a radius of 3 cm and a height of 10 cm. Calculate the area of the sheet required to make its curved surface. Also, describe how this calculation is useful for packaging industries.
Answer:

To find the area of the curved surface of the cylindrical box, we use the formula:


Step 1: Curved Surface Area = 2 × π × radius × height.
Step 2: Substitute the values: 2 × 3.14 × 3 cm × 10 cm.
Step 3: Calculate: 2 × 3.14 × 30 cm² = 188.4 cm².

This calculation is useful for packaging industries because:

  • It helps determine the exact amount of material needed to wrap cylindrical objects.
  • Reduces waste by optimizing material usage.
  • Ensures cost-effective production of packaging.
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