Fold a Net and Check Which Faces Meet

Predict the fold before making the model

A flat arrangement of shapes can look convincing without closing into the solid you expect. A useful way to test a net is to label the faces, predict where they will move, and check whether the edges meet without faces overlapping. This lesson uses two cube nets, one arrangement that fails, and a cuboid whose length, width and height are 4, 3 and 2 units.

Use the diagrams and descriptions for a no-cut reasoning activity. If you choose to make a paper model, ask an adult to prepare and cut the outside boundary with suitable scissors. Keep the shared fold lines uncut. Do not use knives or sharp scoring tools. Gentle paper folds are enough; a model need not be glued or taped shut to explain which faces meet.

Faces, edges and vertices

A cube has six square faces of equal size. A cuboid has six rectangular faces; opposite faces have matching dimensions. Here, adjacent faces means two faces that share an edge when the solid is folded. Opposite faces lie across the inside space from each other and do not share an edge. A vertex is a corner where edges meet.

A closed cube or rectangular cuboid has twelve edges and eight vertices. Do not count all the separate square outlines in a flat net as twelve already joined solid edges. Two free boundary edges can become one edge when the paper folds. Small glue tabs, if an adult adds them, are joining aids, not extra geometric faces.

Worked case 1: predict cube net 1

Text layout: A, B, C and D form a left-to-right row. E is attached immediately above B. F is attached immediately below B. All square sides are equal. “Top”, “bottom”, “left” and “right” in the edge table below refer to the flat diagram as displayed, before folding.

Imagine B as the face nearest you. A and C turn around B’s side edges. D, attached to C, comes around to the face opposite B. E and F close the remaining two sides. This gives the opposite pairs A with C, B with D, and E with F. Every other pair of faces is adjacent in the completed cube.

Notice that A and D do not touch in the flat arrangement, yet they share an edge after folding. Adjacency in the folded solid is therefore not the same as immediate neighbourhood on the page. Predict that A’s left edge will meet D’s right edge; then trace the other joins.

The five edges already connected in the net are A–B, B–C, C–D, B–E and B–F. They become five folded edges. The seven new joins account for the remaining seven: 5 + 7 = 12 solid edges. The word “top” in this table is an edge label on the flat page; two top-labelled edges can meet because their faces rotate in different ways.

Worked case 2: change the attachments

Text layout: A, B, C and D are again in a left-to-right row. E is above A; F is below C. All six squares still have the same side length. The four-square strip wraps around four sides, E closes one end and F closes the other.

This net also gives opposite pairs A–C, B–D and E–F. However, do not reuse net 1’s entire edge-pair table. In net 2, E’s bottom edge is already joined to A’s top edge. E’s right edge meets B’s top edge, E’s top edge meets C’s top edge, and E’s left edge meets D’s top edge. The labels are the same but the flat attachments differ.

A strong prediction names a face or edge and says how it gets there. “Both nets have six squares” is a necessary observation, but it is not enough to prove that either one works. The next arrangement also has six squares.

Worked case 3: diagnose a six-square non-net

Text layout: A, B, C, D, E and F form one uninterrupted left-to-right row. Folding each shared edge through a right angle in the same wrapping direction makes A–D go around four sides. The fifth square, E, comes onto A’s face position; F comes onto B’s face position. Two faces overlap, while the two ends remain open.

That is a specific failure, not simply an unattractive shape. A closed cube needs one face in each of six different positions. Folding a crease the opposite way cannot rescue this straight strip into a cube: it sends a later face back towards a face already used rather than supplying the two missing end faces. Keep this as a prediction-and-reasoning example; there is no need to cut a failed model.

Worked case 4: count the surface of a cuboid

Text layout: the middle strip reads A, B, C, D from left to right. Each face is 2 units high. A and C are 4 units wide; B and D are 3 units wide. E is a 4-by-3 rectangle attached above A along a 4-unit edge. F is another 4-by-3 rectangle attached below A along a 4-unit edge.

The strip wraps around the cuboid’s four side faces. E and F close the remaining faces. Corresponding edge lengths agree: A’s 2-unit side attaches to B’s 2-unit side, while E’s 4-unit edge attaches to A’s 4-unit edge. Opposite pairs are A–C, B–D and E–F.

The whole outside surface is 16 + 12 + 24 = 52 square units. You can check it by adding the individual faces: 8 + 6 + 8 + 6 + 12 + 12 = 52. Count each grid square once. Do not add the area of imagined glue tabs or count both sides of the paper as two outside surfaces.

The space inside is different. One layer of unit cubes on a 4-by-3 base would contain 12 cubes. There are two such layers, giving 12 × 2 = 24 cubic units. Surface area describes covering the outside; volume describes filling the inside. The numbers 52 and 24 answer different questions and require different units.

Three practice tasks with checked answers

Practice 1: neighbours after folding

In cube net 1, which face is opposite B? Are A and D adjacent, even though they are separated on the page?

Answer: D is opposite B. A and D are adjacent: A’s left edge meets D’s right edge. Use the edge join to justify the answer rather than the distance between the labels in the flat drawing.

Practice 2: choose the correct edge join

In cube net 2, does E’s top edge meet C’s top edge or F’s top edge? Explain why the other choice cannot be right.

Answer: E’s top edge meets C’s top edge. E and F close opposite ends of the cube, so they are opposite faces and cannot share an edge. F’s top edge is already attached to C’s bottom edge in this net.

Practice 3: match a measurement to its meaning

A learner writes, “The 4-by-3-by-2 cuboid needs 24 square units of covering because 4 × 3 × 2 = 24.” Repair the answer. How much surface area belongs to A and C together?

Answer: Multiplying the three lengths gives volume, so 24 cubic units is the space inside. Covering all six faces needs 52 square units. A and C each have area 4 × 2 = 8 square units, so together they contribute 16 square units.

A no-cut checking routine

Point to a face and name an attached neighbour. Describe the hinge between them. Imagine the neighbour turning one quarter-turn, then predict one new join. Compare the prediction with the appropriate table or explanation. If a full mental fold is difficult, an adult can hold a prepared paper net while the learner directs one fold at a time.

For a stronger check, trace every free edge and account for its partner. A neat-looking model with a gap or an overlap is not a closed cuboid. A slightly untidy paper model can still demonstrate correct geometry if the intended edges and faces match. Choose the next task from what the learner can explain; no specific current school grade is assumed here.

Frequently asked questions

Is every connected arrangement of six squares a cube net?

No. Six equal squares are necessary, but the arrangement must fold without overlapping faces and close all six face positions. The straight six-square strip fails that test.

What is the difference between adjacent and opposite faces?

Adjacent faces share an edge in the folded solid. Opposite faces lie across the solid’s inside space and share no edge.

Can faces that are far apart in a net become neighbours?

Yes. In cube net 1, A and D are separated on the page but join along A’s left edge and D’s right edge after folding.

Do glue tabs count as faces?

No. They are optional construction aids. Count the six geometric faces for the cube or cuboid, excluding tabs from the stated surface-area calculation.

Why does the cuboid have two faces of each size?

Its opposite faces match. This cuboid has two 4-by-3 faces, two 4-by-2 faces and two 3-by-2 faces.

Why is volume measured in cubic units?

Volume counts three-dimensional unit cubes that fill space. Surface area counts two-dimensional unit squares covering faces.

Must I cut out the nets to answer?

No. Use the labelled layouts, edge joins and explanations, or ask an adult to demonstrate folds on a prepared model. The reasoning task can stay entirely on paper.

Reference: the Centre for Innovation in Mathematics Teaching explanation of surface area describes adding the areas of faces in cube and cuboid nets. All layouts, labels and worked values here are original.