CBSE Class 5 Mathematics NCERT Solutions: Boxes and Sketches

Solution: To determine which shapes can be folded into a box, we need to examine their structure. A shape can be folded into a box if it has a base and four sides that can fold upwards, plus a top that can fold over. Typically, a net for a box consists of 6 squares arranged in a way that allows for folding into a closed rectangular prism.

Let's analyze the typical shapes presented in such problems:

• Shape (a): If this shape has a base and four sides connected, with an additional face that can serve as a top, it can form a box.
• Shape (b): This shape might be too spread out or lack the necessary connections to form a closed box. For instance, if it's a single line of squares, it won't fold into a box.
• Shape (c): This shape is often designed to be a valid net for a box, usually with a base, four sides, and a top.
• Shape (d): Similar to (b), this shape might not have the correct configuration of faces to form a box.

Based on common examples for this exercise, the cutouts that can be folded to make boxes are usually those with a base and four sides that can be folded up, along with a top face. Shapes (a) and (c) are typically designed to be valid nets for a box.

Answer: Cutouts of (a) and (c) can be folded to make boxes.

Solution: A cube is a fundamental three-dimensional geometric shape. It is a regular hexahedron, meaning it has six flat surfaces or faces. Each face of a cube is a perfect square, and all these squares are identical in size. These faces meet at right angles.

Answer: A cube has 6 faces.

Solution: To determine if a 2D shape can fold into a cube, it must be a valid net for a cube. A valid net for a cube is a 2D pattern made of squares that can be folded along the edges to form a cube. Typically, a cube's net consists of 6 squares. There are several possible arrangements of 6 squares that can form a cube. The shapes Buddha drew are usually examples of valid cube nets, meaning they have the correct number of squares (6) and an arrangement that allows them to be folded into a cube without overlapping or leaving gaps.

Answer: Yes, both the shapes Buddha drew can be folded into cubes, assuming they are valid nets for a cube.

Solution: A shape that can fold into a cube is called a net of a cube. A net for a cube is typically made of 6 squares. There are 11 different possible nets for a cube. One common arrangement is a cross shape, where 4 squares are in a line, and one square is attached above and one below the second square in the line. Another valid net could be 3 squares in a column with 2 squares attached to one side of the middle square and 1 square attached to the other side of the middle square.

Here is an example of another shape that can fold into a cube:

Imagine 4 squares arranged in a row. Then, attach one more square directly above the second square in the row, and attach another square directly below the second square in the row. This forms a shape with 6 squares that can be folded into a cube.

(Note: A visual representation would typically be drawn here in a textbook.)

Solution: The problem states that Buddha knows all the faces of a cube are squares. The area of any square is calculated by multiplying the length of its side by itself. Therefore, the area of each face of the cube is equal to the side length of the square face multiplied by the side length of the square face.

The formula for the area of a square is: Area = Side × Side.

Answer: Area of each face of the cube = Side × Side.

Solution: A shape that does not fold into a cube is one that is not a valid net for a cube. This can happen for several reasons: it might not have exactly 6 squares, or the 6 squares might be arranged in a way that they cannot be folded to form a closed cube without gaps or overlaps. For example, a shape made of 6 squares arranged in a 2x3 rectangle cannot be folded into a cube. Another example is a shape with fewer or more than 6 squares.

(Note: A visual representation of such a shape would typically be drawn here in a textbook.)

Answer: A shape with 6 squares arranged in a 2x3 rectangular block will not fold into a cube.

Solution: A cube is a 3D shape with six equal square faces. Many everyday objects are designed in the shape of a cube or closely resemble one. Identifying these objects helps in understanding the concept of a cube in a practical context.

Examples of objects that look like a cube include:

• Dice used in games
• Chalk boxes
• Rubik's cubes
• Some small gift boxes
• Sugar cubes
• Some building blocks

Answer: Following objects look like a cube: Dice, chalk box, Rubik cube, etc.

Solution: An open box is a box without a lid or top face. To form an open box, the net needs to have 5 faces: a base and four sides. The shapes that can be folded into an open box will typically consist of 5 squares arranged in a way that allows them to form the base and the four walls of the box. Common nets for open boxes include a row of 4 squares with one square attached above or below one of the inner squares, or a T-shape arrangement of squares.

By examining the 8 shapes provided in the textbook, we can identify those that have exactly 5 squares arranged in a suitable pattern. These are the shapes that can be folded to create an open box.

(Note: The specific shapes that fold into an open box would be indicated by referring to the textbook's diagrams, typically shapes numbered 1, 2, 3, 5, 6, 7, 8 are often shown as valid for open boxes, while shape 4 might not be.)

Answer: The 8 shapes which can be folded to make an open box are typically indicated by the textbook's diagrams. Based on common representations, shapes like 1, 2, 3, 5, 6, 7, and 8 are usually shown as valid nets for an open box.

Solution: Shapes that cannot be folded to make an open box are those that do not form a valid net for an open box. This could be because they have too many or too few squares, or the squares are arranged in a way that prevents them from forming the base and four sides of an open box. For instance, a shape with 6 squares cannot form an open box (it might form a closed box or not fold correctly). A shape with only 4 squares arranged in a line cannot form an open box as it lacks enough faces.

Examples of shapes that will not fold into an open box:

• A shape made of 6 squares arranged in a 2x3 rectangle.
• A shape made of 4 squares in a straight line.
• A shape with squares arranged in a way that leaves gaps when folded or causes overlaps.

(Note: A visual representation would typically be drawn here in a textbook.)