Solid nets

1Summary 重點摘要

Unfold a solid flat to see all faces.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What unfolding meansCut a juice carton along its edges and lay it flat: the solid becomes one joined flat pattern, the process is called unfolding, and the flat pattern is the net. Why learn nets? Because laid flat the shape is easy to measure and calculate, folded up it becomes the solid again, and nets reveal the structure of solids; packaging design starts from nets too. Every solid has its own nets, and cutting along different edges can give different patterns. A common misunderstanding is thinking each solid has only one net; a cube alone has more than a dozen layouts, and any that folds back is correct.
2. Step 2: The cube netA cube unfolds into 6 equal squares, arranged as a cross, a 1-4-1 row and more than ten other styles. Although the styles are many, every true one folds back into a cube, because all 6 faces are identical and any arrangement that closes up is fine. To judge a pattern, fold it in your mind and watch whether the 6 faces would overlap; for instance, five squares in one long row must overlap when folded, so it is not a net. A common error is counting 6 faces and judging it correct without checking overlap; both the count and the fold must pass for a reliable judgement.
3. Step 3: The triangular prism netA triangular prism has 2 triangular bases and 3 rectangular sides, so its net has 5 faces: 3 rectangles joined in a row with 1 triangle on each side. Why this combination? When folded, the 3 rectangles wrap the body of the prism and the 2 triangles cap the top and bottom, closing the solid completely. The most common layout is three rectangles in the middle with a triangle on each side. A common error is drawing both triangles on the same side of the row; then they crowd onto one end when folded and the other end stays open, which a mental fold test reveals.
4. Step 4: The square pyramid netA square pyramid has a square base and 4 triangular sides, so its net has 5 faces: 1 square in the middle with 4 triangles joined around it. When folded, the 4 triangles lean inward and their tips meet at one point, the apex. Why are the sides triangles? Because each side of a pyramid narrows from a base edge up to the apex, so every side face must be a triangle; the Great Pyramid is exactly this shape. A common error is drawing rectangular sides, which belong to prisms, or drawing only 3 triangles so one base edge stays uncovered; counting the side faces catches both.
5. Step 5: The cylinder netA cylinder unfolds into 3 faces: the side unrolls into one rectangle, with 1 circle joined above and 1 below. The rectangle's width equals the cylinder's height, and its length equals the circumference of the base circle, which is the only way the rectangle can wrap the two circles exactly. Why is the unrolled side a rectangle? Roll a sheet of paper into a tube and unroll it once: a curved surface flattened must become a rectangle. A common error is writing the length as the diameter instead of the circumference; remember the length is 2 × π × radius, about 12.56 cm for radius 2 cm.
6. Step 6: The fold testTo judge whether a pattern is the net of a solid, the best method is to fold it in your mind: first count whether the faces are enough, then watch for overlaps, and finally check for openings. Take the triangular prism: 3 rectangles wrap into a ring, 2 triangles cap the ends, and with both the face count and the shapes correct, the net is right. Why is this test reliable? Because a net must satisfy three conditions at once: the right number of faces, no overlap, and no gap; missing any one disqualifies it. The common error is checking only one condition, such as the count alone, while two faces overlap and one end is left uncovered.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A net is the flat pattern of a cut-open solid
• Cube: 6 squares; triangular prism: 5 faces (2 triangles + 3 rectangles)
• Square pyramid: 1 square + 4 triangles
• Cylinder: 1 rectangle + 2 circles
• Real nets pass the fold test: no overlaps, no gaps