Nets of solids
Hand someone a flattened cardboard box and they can usually fold it back into a box without thinking twice — six rectangles hinged along the right edges, ready to close up into a familiar shape. That flattened pattern is a net: a 2-D layout of a solid's faces that, folded along its internal edges, reconstructs the closed 3-D solid exactly, with every face present once and every edge meeting its partner. Nets sit at the heart of the ICSE Class 8 chapter "Representing 3-D in 2-D." The skill is twofold: given a solid, draw a net that will fold into it; and given a flat pattern, decide whether it will fold into a valid solid or whether it leaves a gap or forces an overlap. The topic is deliberately restricted to solids with flat polygonal faces — cubes, cuboids, prisms and pyramids. Curved solids (cylinders, cones, spheres) and their curved-surface nets belong to Class 10 mensuration and are not covered here.
A net is a valid, gap-free, overlap-free flat arrangement of all of a solid's faces, connected edge-to-edge so that folding along every internal line closes every edge of the solid exactly once.
The core idea
What makes a net valid. Two conditions must both hold. First, the net must contain exactly the right faces — matching every face of the solid in shape and size, with none missing and none extra. Second, the arrangement of those faces must be foldable: when every internal edge is folded, the boundary edges of the solid must meet up cleanly, with no gap left open and no two faces forced to overlap. Having the correct pieces is necessary but not sufficient — the layout matters just as much as the inventory of shapes. A useful way to see why arrangement matters is to count. A solid with faces, edges and vertices satisfies Euler's relation:
where: is the number of faces, the number of edges, the number of vertices. A cube has , , , and indeed . This relation is a bookkeeping check on the solid itself, not on the net — but it is a reminder that every one of the solid's edges must be accounted for exactly once when the net's flat edges are folded and matched together. If a proposed net has too few internal fold lines, or the wrong faces glued along the wrong edges, some of those meetings never happen and the shape will not close.
Cube net. Six identical squares, arranged so every fold brings all six together into a closed box with no gap or overlap. Several different arrangements work — for example, four squares in a row with one extra square attached above and one below (a cross-like shape), or four in a row with one square on each side. Not every arrangement of six squares works: some layouts leave a face missing a partner edge, so folding leaves a gap or forces two faces to overlap instead of meeting cleanly.
Cuboid net. Three pairs of identical rectangles, since a cuboid's opposite faces are always equal in shape and size. The pairs must be arranged so that after folding, each pair ends up on opposite sides of the box — not adjacent to each other in the final solid.
Prism net. Two identical polygon-shaped ends (matching the prism's cross-section) plus a set of rectangles equal in number to the sides of that polygon, joined edge-to-edge in a strip so the rectangles wrap around and connect the two polygon ends.
Pyramid net. One polygon base plus one triangle for every side of the base, each triangle attached along a full edge of the base (not merely touching at a corner), so that folding brings every triangle's outer tip up to meet at a single apex point.
The goal is to show, using nothing beyond the six-rectangle inventory of a cuboid's net, that adding the areas of all six flat pieces gives the same single formula every time — — regardless of which valid arrangement of the six rectangles you draw, because folding never creates or destroys area, it only bends the flat sheet along its internal edges.
Step 1. Identify the three distinct rectangle sizes a cuboid net must contain, matching its three pairs of opposite faces: a pair (top and bottom), a pair (front and back), and a pair (left and right sides), where , , are the cuboid's length, breadth and height.
Step 2. Write the area of one rectangle from each pair: , , and . Because each of the three sizes appears exactly twice in the net (once for each face in its opposite pair), the total area contributed by all six pieces is twice the sum of these three: .
Step 3. Confirm this sum does not depend on the arrangement. Whether the six rectangles are laid out in a cross-like pattern, a row with pieces above and below, or any other valid foldable layout, the set of six pieces is the same six rectangles with the same six areas — only their position in the flat plane changes, never their size. Folding along the internal edges only bends the sheet; it adds no extra area and removes none, so the flat net's total area before folding equals the solid's total surface area after folding.
Step 4. State the result. Summing the six pieces gives the total surface area of the cuboid directly from its net:
where: , , are the cuboid's length, breadth and height, and is the combined area of all six net pieces, equal to the cuboid's total surface area.
Worked example
Problem. A cuboid has length 4 cm, breadth 3 cm and height 2 cm. Describe a valid net for it and find the total area of the net (equal to the cuboid's total surface area).
Step 1. List the three pairs of rectangular faces a cuboid must have. Opposite faces are equal, so there are exactly three distinct rectangle sizes, each appearing twice: (top and bottom), (front and back), and (left and right sides).
Step 2. Arrange the six rectangles so every adjoining pair in the flat layout shares a common edge of matching length. One valid net places the four side faces — , , , — in a row, then attaches the two rectangles above and below one of the rectangles in that row. Every shared edge in this layout has a matching length on both sides, so folding along each internal line closes the box with no gap.
Step 3. Add the six face areas. Since opposite faces repeat, this is times the sum of the three distinct face areas:
where: cm, cm, cm are the cuboid's length, breadth and height, and is the total surface area, which equals the total area of the flattened net.
Answer. The net is 3 pairs of rectangles (, , cm), arranged edge-to-edge as in Step 2, with total area .
Competitive stretch
For a square pyramid with base side 5 cm and 4 identical triangular faces, the net is drawn by placing the 5 cm square base in the centre and attaching one triangle to each of its four sides, each triangle's base edge matching the square's edge exactly. The net has 5 pieces total — 1 square base and 4 congruent triangles — and folding brings all four triangle tips together at a single apex above the centre of the square.
Common traps
- Assuming any collection of the right face shapes automatically folds into the solid — the arrangement, not just the inventory of shapes, decides whether it closes without gaps or overlaps.
- Pairing adjacent faces instead of opposite faces when sizing a cuboid net — opposite faces are equal, adjacent ones generally are not.
- Attaching a pyramid's triangular faces so they only touch the base at a corner instead of sharing a full edge, which removes the hinge line needed to fold up to the apex.
- Drawing a net with the wrong piece count, such as a pyramid missing one triangle or a prism missing one rectangular side face.
Ask the tutor: If you have a specific flat pattern and are unsure whether it folds into a cube, cuboid, prism or pyramid, describe it to the AI tutor and ask it to trace the fold with you step by step.
How do I know if a given flat pattern will actually fold into a cube, or if it will leave a gap
Trace the fold mentally, edge by edge, one square at a time from a fixed square you call the base. Fold each neighbouring square up along their shared edge and track where its far edges land. A pattern of six squares works only if, after all folds, every square ends up covering exactly one face with no two squares landing on the same face and no face left uncovered. If you get stuck, describe the exact arrangement of squares to the AI tutor and ask it to walk through the fold step by step with you.
Do all six squares of a cube net have to be arranged in one connected strip
No — a strip is only one style. Any of several different connected arrangements of six squares can fold into a cube, such as a cross shape (four in a row with one above and one below) or an arrangement with squares on two different sides of the row. What matters is not the specific layout but that every square ends up sharing a fold edge that lets it close correctly against its neighbours with no gap or overlap. Not every six-square arrangement works, so each one has to be checked by tracing the fold.
Why don't cylinders and cones get a net in this ICSE Class 8 chapter
Because their curved surfaces cannot be represented by flat polygon pieces the way cube, cuboid, prism and pyramid faces can. Unrolling a cylinder's curved surface or a cone's slanted surface into a flat shape involves curved-surface area formulas that use the circle's circumference and radius, which is mensuration content taught later, in Class 10. This chapter deliberately limits nets to solids whose every face is already flat, so cube, cuboid, prism and pyramid are the complete list here.