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Nets of cubes and drawing 3D shapes

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visibility 251update 4 months agobookmarkshare

🎯 In this topic you will

Identify and draw the different nets that form open and closed cubes.
Identify, describe, and draw 3D shapes, including using isometric paper.

🧠 Key Words

cube
open cube

Show Definitions

cube: A 3D shape with six equal square faces, twelve equal edges, and eight vertices.
open cube: A cube net or structure where one or more faces are missing, so the shape is not fully enclosed.

Cubes in Real-World Design

C ubes and open cubes are important shapes in design and technology. Many everyday objects are created using cube-based structures because they are strong, simple, and easy to build.

W here do you see things designed with cubes?

📘 Worked example

Draw this cube on isometric paper.

Step 1. Make sure that your isometric paper is positioned correctly.

Step 2. Draw a dot representing one vertex of the cube.

Step 3. Draw lines that represent the edges of the cube that connect to that vertex.

Step 4. Draw a single face of the cube.

Step 5. Complete all the other visible faces of the cube.

Answer:

Step 1. The isometric grid must be aligned correctly so that all cube edges follow the dot pattern.

Step 2. The first dot marks a single vertex and sets the position of the cube.

Step 3. Draw three edges meeting at the vertex, each following one of the isometric directions.

Step 4. Use the edges to form one complete face of the cube.

Step 5. Add the remaining faces using parallel lines to complete the cube.

❓ EXERCISES

$1.$

a. How many faces does a cube have?

b. Describe the shape of the faces of a cube.

👀 Show answer

a. A cube has $6$ faces.
b. All faces are squares of equal size.

$2.$ What $3$D shape will this net make?

👀 Show answer

The net forms a cube.

$3.$ Which of these nets will not make an open cube?

👀 Show answer

Nets **C** and **E** do not form an open cube. Their arrangements cannot fold properly to leave an open cube.

$4.$ Draw a net of a cube. Trace this square to use as a template for each face of your cube.

👀 Show answer

Many correct cube nets exist. A valid net contains $6$ joined squares that can fold to form a cube.

🧠 Reasoning Tip

Think about what parts of a $3$D shape you can see and what parts you cannot see.

$5.$ The diagram shows what Sofia can see when she looks at two different $3$D shapes.

a. What two shapes could they be?

b. Explain how you know they could be those shapes.

c. Sketch each of the shapes in a different orientation. Label each shape with its name.

👀 Show answer

a. The larger circle could be the base of a cylinder or a sphere seen head-on; the smaller circle could be the base of a smaller cylinder or a smaller sphere.
b. A circular face is seen when looking directly at the end of a cylinder, or when viewing a sphere straight on.
c. Sketches will vary; correct labelled sketches of a cylinder and a sphere are acceptable.

$6.$ Find a triangular prism. Put the triangular prism on a flat surface and look at it from one position. Draw the triangular prism from your position.

a. How many triangles can be seen in your picture?

b. How many rectangles can be seen in your picture?

👀 Show answer

Answers depend on orientation. Typically you can see $1$ triangular face and $2$ rectangular faces from a fixed viewpoint.

$7.$ Draw these models on isometric paper.

👀 Show answer

Drawings will vary. Each model should be represented accurately on isometric paper.

$8.$ Make cuboid b, c, or d shown in question $7$ using cubes. Turn the cuboid so that it sits on a different face. Draw the cuboid in its new position on isometric paper.

👀 Show answer

Answers will vary. A correct drawing shows the chosen cuboid resting on a different face while preserving the same dimensions.

🧠 Think like a Mathematician

The net of a cube has been cut into these two pieces:

Imagine the two pieces were stuck together with sticky tape. Draw two sets of nets that can be made by the two pieces stuck together. One set is the nets that can be folded to make a cube. The other set is the nets that cannot be folded to make a cube.

Example:

Test your nets by copying them onto squared paper. Cut out and fold the nets to see if they are in the correct set.

You are specialising when you choose and test a net to see if it will make a cube.
You are classifying when you put each net into a set.

Task (solo version): Use the two given pieces to construct as many different nets as you can. Sort them into:

Nets that will fold to make a cube
Nets that will not fold to make a cube

Follow-up Questions:

1. How do you decide whether a net will fold to make a cube?

2. Why do some arrangements fail to make a cube even though they contain six squares?

3. Explain how the positions of the squares affect whether the net can fold properly.

Show Answers

1: A net will fold to make a cube if every square can become one of the six faces without overlapping and if the arrangement allows three faces to meet at each vertex. Visualising or physically folding helps confirm this.
2: Some arrangements fail because the squares are positioned so that faces collide or cannot rotate into the correct orientation. Even with six squares, the joining edges may prevent a full enclosure.
3: For a net to work, squares must be arranged so that each face has the correct neighbours. If a face is placed in an impossible location (e.g., diagonal without proper edge attachment), folding becomes impossible.

💡 Quick Math Tip

Use net testing to check cube validity: When deciding if a net can fold into a cube, imagine lifting and folding each square. A correct cube net allows all six faces to fold together without overlap, with exactly three faces meeting at each vertex.

Deep Dive Questions

Use these questions to deepen understanding and apply ideas in new ways. Explain your reasoning clearly.

CONCEPTUALQ1. Why must a cube net contain exactly $6$ faces, and why must each face be a square?

Show answer

A cube has $6$ congruent square faces in three perpendicular pairs. Any valid net must include all faces so that the solid can fully enclose space. If one or more faces were missing, the shape would not close; if the faces were not squares, the resulting solid would not be a cube. A common misconception is thinking that “as long as there are six shapes” the net will work — but those shapes must be identical squares.

ERROR ANALYSISQ2. A student claims that any arrangement of $6$ squares touching edge-to-edge is a cube net. Identify the flaw in their reasoning and explain how you can show a net does *not* fold into a cube.

Show answer

The flaw is assuming that the *number* of squares is the only requirement. The *positions* of those squares determine whether they can rotate around their shared edges without overlapping. For example, if three squares meet in a straight line with additional faces branching incorrectly, some faces will collide when folded. To show a net fails, imagine folding each face around its edge; if any faces overlap or leave no place for a final face, it is not a valid cube net.

TRANSFERQ3. A triangular prism has $2$ triangular faces and $3$ rectangular faces. Explain how its net differs from a cube net and how knowing cube nets helps you interpret nets of other solids.

Show answer

A prism net includes rectangles arranged in a chain with matching triangular faces attached at two ends. Unlike a cube net, the faces are not identical, so the arrangement must reflect the structure of the solid: the triangles must align so they form parallel surfaces, and the rectangles must wrap around the side. Understanding cube nets trains you to check adjacency and folding pathways — skills transferable to reasoning about any polyhedron net.

MULTI-STEPQ4. You are given two disconnected cube-net pieces (as in the lesson). Determine whether *every* way of taping the pieces together edge-to-edge could result in a valid cube net. Justify your conclusion.

Show answer

Not every arrangement will work. Even if each piece is made of squares, some attachments create branching shapes where a face blocks the folding path of another. A valid cube net must allow a sequence of rotations in which no face overlaps and all six faces enclose space. When the two pieces are taped in positions that misalign future folds — for example, attaching a square on a “corner” that requires incompatible rotations — the net becomes impossible. A key step is checking adjacency: each face must border the correct neighbours of a cube, and the degree of each vertex must not exceed three faces.

MINI INVESTIGATIONQ5. Explore whether there exists a cube net where the six faces can be folded in more than one distinct sequence to form the cube. Provide an argument or example supporting your conclusion.

Show answer

Yes — many cube nets allow multiple folding sequences. For example, a “strip of four squares with two attached opposite” can be folded by raising different side faces first. The adjacency constraints are fixed, but the order in which faces are rotated is flexible as long as the final structure closes correctly. A common mistake is assuming one unique fold order; in reality, the cube has symmetrical structure, so several folding paths can lead to the same 3D result.

📘 What we've learned

We learned how to recognise nets of cubes, including full cube nets and open cube nets.
We practiced identifying which arrangements of $6$ squares can fold to form a cube and which cannot.
We learned how to draw cube nets and use them to visualise three-dimensional structures.
We explored how to interpret $3$D shapes from different orientations and understand which faces are visible from various viewpoints.
We developed skills for drawing $3$D shapes on isometric paper, beginning from a single vertex and constructing all edges correctly.
We used mathematical reasoning to explain why certain nets fail and how folding paths help determine valid cube nets.

Nets of cubes and drawing 3D shapes

🎯 In this topic you will

🧠 Key Words

Cubes in Real-World Design

❓ EXERCISES

🧠 Reasoning Tip

🧠 Think like a Mathematician

💡 Quick Math Tip

Deep Dive Questions

📘 What we've learned

Related Past Papers

Related Tutorials

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