Net of cuboid

1Summary 重點摘要

Cuboid unfolds into 6 rectangles.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What a net isCut a tissue box along its edges and lay it flat: the joined flat pattern is the net of the cuboid. The net shows clearly what shape every face is and how the faces connect; folding along the lines rebuilds the original box. Why learn the cuboid's net? Because it is the basis of computing surface area and of understanding solid structure, and packaging design begins from nets. A common misunderstanding is thinking there is only one net; in fact the same cuboid can be laid flat into several different patterns, and every one that folds back is correct.
2. Step 2: Six faces in three pairsA cuboid has 6 faces in 3 pairs. For a 5 × 4 × 3 box: the top and bottom faces are 5 × 4, the front and back are 5 × 3, and the left and right are 4 × 3. Opposite faces are exactly the same size, an essential feature of the cuboid that shows equally clearly in the net. Why three pairs? Because the three directions of length, width and height each correspond to one pair of opposite faces, and 6 faces form exactly three pairs. A common error is treating all 6 faces as different, or pairing wrongly, such as matching 5 × 3 with 4 × 3; check pair by pair using length, width and height and the pairing stays ordered.
3. Step 3: Many layouts workThe same cuboid can be laid flat into a cross shape, a T shape and other patterns; they look different, but all are 6 rectangles joined together. Any layout that folds back into the original box is correct. Why more than one layout? Because the cut edges may differ, the edges left joined differ, and the flattened shape differs accordingly. So judging a net cannot depend on whether the shape looks familiar; one must think about the result after folding. A common error is accepting only the cross shape and rejecting the rest; the true standard is always whether it folds back into the original solid, never the appearance of the pattern.
4. Step 4: Spotting fakesSome patterns are not cuboid nets: for example, in one layout two 5 × 4 faces would fold on top of each other, or a face is missing, or an extra face appears; none of these can build the box. The judging method is to fold once in the mind, checking one by one for overlaps and for missing faces. Why do overlaps happen? Because two faces have been placed where only one can sit, and the cuboid has just one pair of faces in each direction. A common error is counting 6 faces and approving at once, yet 6 faces can still overlap; the count, the positions and the pairing all need checking, with none skipped.
5. Step 5: Link to surface areaLaid flat, the sum of all the faces' areas is the surface area. For the 5 × 4 × 3 box: 5 × 4 = 20, 5 × 3 = 15, and 4 × 3 = 12, so the surface area = 20 × 2 + 15 × 2 + 12 × 2 = 40 + 30 + 24 = 94 cm². Computing face by face on the net leaves no face behind. Why does the net help? Because every face lies open before the eyes once flattened, so each can be computed directly and then added, with the picture and the arithmetic confirming each other. A common error is missing one pair and adding only 4 faces; remember that a cuboid always has 6 faces in three pairs.
6. Step 6: Count the facesA cuboid's net must have exactly 6 rectangles. Count the faces: fewer or more than 6 is always wrong, and this is the quickest first check of a net. After counting, see whether the faces appear in equal pairs of the same size, and finally fold once in the mind to confirm there is no overlap. Why three checks? Because each step rules out one class of error: the count removes missing or extra faces, the pairing removes mismatched sizes, and the fold removes overlapping positions. A common error is stopping after the first step; only when all three pass can the net be judged correct.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A cuboid net = six joined rectangles
• Opposite faces match: top/bottom, front/back, left/right
• One cuboid can have several different nets
• Layouts that overlap when folded are not nets
• All faces of the net add up to the surface area