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Geometry - Nets of 3D shapes

Grade 4ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A net is a 2D flat shape that can be folded to make a 3D solid. Imagine unfolding a cardboard box; the flattened shape you see is the net of that box.

A net of a cube consisting of six squares arranged in a cross shape.
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A net for a cuboid consists of 66 rectangles. In opposite pairs, these rectangles must be identical to form the parallel faces of the 3D shape.

Net of a cuboid showing different sized rectangular faces.
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A triangular prism net is made of 22 triangles (the bases) and 33 rectangles (the lateral faces). The sides of the triangles must match the widths of the rectangles.

Net of a triangular prism showing three rectangles in a row and two triangles on the sides.
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A square pyramid net has one square base in the center and four triangles attached to its sides which meet at a single point when folded.

📐Formulae

Number of faces in a Cube=6\text{Number of faces in a Cube} = 6

Number of faces in a Cuboid=6\text{Number of faces in a Cuboid} = 6

Number of faces in a Square Pyramid=1 (base)+4 (triangles)=5\text{Number of faces in a Square Pyramid} = 1 \text{ (base)} + 4 \text{ (triangles)} = 5

Number of faces in a Triangular Prism=2 (triangles)+3 (rectangles)=5\text{Number of faces in a Triangular Prism} = 2 \text{ (triangles)} + 3 \text{ (rectangles)} = 5

Euler’s Formula: V+F−E=2 (where V=Vertices, F=Faces, E=Edges)\text{Euler's Formula: } V + F - E = 2 \text{ (where } V=\text{Vertices, } F=\text{Faces, } E=\text{Edges)}

Surface Area of a Cube=6×(side×side)\text{Surface Area of a Cube} = 6 \times (\text{side} \times \text{side})

💡Examples

Problem 1:

A net is made up of 22 identical circles and 11 rectangle. Which 3D shape will be formed when this net is folded?

Solution:

Step 1: Identify the components of the net. We have 22 circles and 11 rectangle. Step 2: Recall the properties of 3D shapes. A shape with two circular bases and a curved surface (which unfolds into a rectangle) is a cylinder. Step 3: Therefore, the shape is a Cylinder.

Explanation:

In a cylinder net, the two circles represent the top and bottom faces, while the rectangle represents the curved side that wraps around the circles.

Problem 2:

Can a net with 55 squares be used to fold a complete cube? Why or why not?

Solution:

Step 1: Count the number of faces in the given net. The count is 55. Step 2: Determine how many faces a cube has. A cube has 66 faces. Step 3: Compare the numbers. Since 5<65 < 6, the net is missing one face. Conclusion: No, it cannot form a complete cube.

Explanation:

A complete cube requires 66 square faces to close all sides. A net with only 55 squares would result in an 'open' box with one side missing.

Problem 3:

Identify the 3D shape that can be formed from a net consisting of one square and four equilateral triangles attached to its sides.

A net of a square pyramid with a central square and four triangles.

Solution:

The shape is a Square Pyramid.

Explanation:

A square pyramid has 55 faces: 11 square base and 44 triangular faces that meet at a vertex. The net described perfectly matches these components.

Problem 4:

Calculate the total surface area of a cube if its net is made of six squares, each with a side length of 5 cm5\text{ cm}.

A single square face from a net with side labeled 5 cm.

Solution:

Area of one square=5×5=25 cm2Area \text{ of one square} = 5 \times 5 = 25 \text{ cm}^2 Total Surface Area=6×25=150 cm2Total \text{ Surface Area} = 6 \times 25 = 150 \text{ cm}^2

Explanation:

A cube has 66 identical faces. Since the net is made of 66 squares of side 5 cm5\text{ cm} each, we find the area of one face and multiply by 66.