Nets: which flat shapes fold into which solid

What to assume they know

• Constructing a solid in your head from a description — profile, outline and viewpoint, and the habit of operating on an imagined shape
• Names of cube, cuboid, cylinder, cone, prism and pyramid from earlier classes
• Square, rectangle, triangle, circle, and equilateral triangle as plane shapes
• Perimeter and circumference; that a circle of radius r has circumference 2πr
• Congruence, at the level of "these two shapes are identical"
• Rotation and reflection of a plane figure, enough to decide when two flat pieces are the same piece
• Counting arguments of the "n of these, each with three of those" kind

What they should be able to do

• Define a net as the surface of a solid unfolded onto a plane, and say what the definition excludes
• Identify faces, edges and vertices of a given solid and count each
• State the general definition of a prism and of a pyramid, and derive the face, edge and vertex counts for an n-sided base
• Decide whether a given flat arrangement of squares folds into a cube, and give a reason for each rejection
• Explain what "two nets are the same" means when rotation and reflection are allowed, and why the count 11 depends on that convention
• Construct the net of a cylinder and state the dimensions of its rectangle
• Explain why unrolling a cone gives part of a circle, using the fact that the apex is equidistant from every point of the base circle
• Explain why a sphere has no net, in terms of what folding can and cannot do
• Distinguish the net proper from the flaps added to make a physical model

Where it usually goes wrong

• "Six squares joined edge to edge always fold into a cube." Candidates (i) and (v) on Part II pp.80–81 have exactly six squares each and neither folds. The count is necessary and nowhere near sufficient.
• "A candidate fails because a face is missing." It fails because two faces land in the same place, which leaves a different face missing as a consequence. Name the overlap, not the gap — that is the diagnostic that generalises.
• "There is one net per solid." A cube has 11, an octahedron 11, a dodecahedron 43,380, a regular tetrahedron 2. The chapter prints all four numbers.
• "A net includes the flaps." The chapter says explicitly that it does not. The flaps are a manufacturing aid; the net is the unfolded surface.
• "11 is just a fact to memorise." It is a fact relative to a convention: two arrangements are counted as one if a rotation or a flip relates them. Change the convention and the number changes. The chapter prints three arrangements side by side precisely to fix the convention before stating the number.
• "The cylinder's rectangle has the diameter as its long side." It has the circumference, 2πr, which is over three times the diameter. This is the single most common numerical error in cylinder nets.
• "Unrolling a cone gives a triangle." It gives a sector — a piece of a disc. The curved edge is forced: every point of the base circle is at distance l from the apex, so in the flattened piece they all sit on a circle of radius l.
• "A sphere has a net, we just haven't found it." It has none, and the reason is structural rather than a failure of ingenuity. Say the reason: folding does not stretch.
• "Prism and pyramid are just names for particular shapes." They are general definitions with a parameter. Once you can count faces, edges and vertices for the n-sided case, the 10-sided case needs no picture — which is the point of the chapter's two questions on Part II p.79.

Questions to check understanding

• Count faces, edges and vertices of a named solid, and of a prism or pyramid with an n-sided base
• Decide whether a given arrangement of six squares is a net of a cube, with a reason for a rejection
• Complete a partially drawn net so that it folds into a stated solid
• Draw, with measurements, a net of a cuboid with stated sidelengths, and check it against the total surface area
• State the dimensions of the rectangle in a cylinder's net for a given radius and height
• Explain why unrolling a cone produces part of a circle
• Explain why no net of a sphere exists
• Match nets to solids in a multiple-choice grid, which is the form the board most often uses on this material

Examples worth working on the board

The chapter prints no answers to any exercise item; everything marked not in the book is worked out here on the chapter's stated inputs.

• The solids figure (Part II p.78, six coloured drawings in two rows, captioned): cuboid, parallelopiped, cylinder; then cone, triangular prism, triangular pyramid. This is the chapter's inventory of what can be made from a foldable flat sheet, and Part II p.77 adds that this is one of the ways hollow solids are actually manufactured.
• Faces, edges, vertices (Part II p.78, second figure: two rows of three line drawings, a cuboid above and a triangular pyramid below, with Vertex, Edge and Face labelled by arrows and the face indicated by hatching). Definitions as printed: faces are the flat surfaces bounding the solid; edges are the segments forming the sides of the faces; vertices are the points where edges meet. The worked instance the page states is the cuboid, and equally the cube: 6 faces, 12 edges, 8 vertices.
• Prisms and pyramids (Part II p.79, two figures of three each: triangular, pentagonal and hexagonal prisms above; triangular, pentagonal and hexagonal pyramids below). Definitions as printed, plus the naming convention and the note that a triangular pyramid is also called a tetrahedron.
• The two counting questions (Part II p.79). Suppose a prism's pair of congruent end polygons are ten-sided — what are its counts of faces, of edges and of vertices? And what are they for an n-sided pair? The same two questions are then asked of a pyramid, first for a ten-sided base and then for an n-sided one. Not in the book: an n-sided prism has n + 2 faces (two bases plus n side faces), 3n edges (n on each base plus n joining them) and 2n vertices, so a 10-sided prism has 12 faces, 30 edges, 20 vertices. An n-sided pyramid has n + 1 faces, 2n edges (n on the base plus n to the apex) and n + 1 vertices, so a 10-sided base gives 11 faces, 20 edges, 11 vertices. Check both against the cube's stated 6-12-8 by putting n = 4 in the prism formulas: 6, 12, 8.
• Fig. 4.1 (Part II p.80). A net of a cube, printed at the top right of the page as an outline drawing. An added reading of the figure: six squares arranged as a vertical strip of four with one square attached to the left and one to the right of the third square in the strip. The reader is asked to visualise the fold.
• Practical aspects (Part II p.80). Two requirements are named: the material must be sturdy enough for the finished cube to stand, and there must be some way of attaching faces to each other — cello tape for cardboard, or extra flaps for weaker material such as chart paper, as seen on packaging boxes. Three artwork panels labelled (i), (ii), (iii) show a flapped orange net, a hand folding it up, and the finished cube. The chapter then states that the net of a solid means the unfolded shape only, and not the supporting flaps.
• The six candidates (Part II pp.80–81, Figure it Out, item 1: which of these are nets of a cube — answer by visualisation first, then with cut-outs). Each candidate has six unit squares. Added descriptions, read from the printed page: (i) a row of four squares with a two-square domino attached above its left end, extending one column beyond it; (ii) two rows of three, overlapping in exactly one column at the ends; (iii) a staircase of three dominoes, each stepped one column right of the one above; (iv) a row of four with one square above and one square below the same, third, column; (v) a row of four with two squares stacked below its second column; (vi) a row of four with one square above the third column and one below the second.
• An added classification, with reasons — the chapter gives none. Nets: (ii), (iii), (iv), (vi). Not nets: (i) and (v). The reasons are the substance of section 7. For (iv) and (vi): a row of four wraps into a band of four side faces, and any one square above plus any one square below supplies the top and the bottom, so every arrangement of that kind folds — six of the eleven nets are of exactly this form. For (ii): take the lower row of three as front, right and back; the square above the front becomes the top; the square to its left becomes the left face; and the last square folds down as the bottom — six distinct faces, no clash. For (iii): the staircase folds a face at a time around the cube and closes. For (i): the extra square attached beyond the top square folds onto a face of the band that is already occupied. For (v): the second square of the hanging pair folds up onto the band as well. An overlap, not a shortage, is what kills a candidate — both failures have exactly six squares.
• Item 2 (Part II p.81, marked Try This). A cube has 11 possible nets in total, where two are counted the same if a rotation or a flip carries one to the other; three such equivalent nets are printed as an example. Find all 11. Not in the book: the 11 fall into four families — six of the row-of-four kind, three in which the six squares form rows of two, three and one, one staircase, and one made of two rows of three. That grouping is not printed and is offered so an explanation can show the 11 in an order that means something rather than as a jumble.
• Item 3 (Part II p.81). Two cuboids are to have nets drawn for them. The first has edges of 5 cm, 3 cm and 1 cm; the second has edges of 6 cm, 3 cm and 2 cm. Not in the book: the six faces come in three congruent pairs, so for (i) the areas are 5 × 3, 5 × 1 and 3 × 1 twice over, giving a total surface area of 2(15 + 5 + 3) = 46 cm²; for (ii), 2(18 + 12 + 6) = 72 cm². A drawn net that does not total to that is wrong, which gives the class a check on its own drawing.
• The regular tetrahedron (Part II p.81). Defined as a tetrahedron with equilateral faces. Four candidate arrangements of triangles are printed and the reader is asked which are nets; the chapter then states that a regular tetrahedron has only 2 possible nets. An added reading of the four candidates, from the printed page: the first is a large triangle divided into four by joining the midpoints; the second is a strip of three triangles forming a trapezium with a fourth triangle hanging below its left end; the third is four triangles in a strip forming a parallelogram of base two and slant one; the fourth is a parallelogram strip of four with a fifth triangle hanging below its left end. An added classification: the two nets are the first and the third. The second is neither of them, and the fourth has five faces for a four-faced solid.
• Two construction tasks (Part II p.81, marked Math Talk). Draw a net with suitable measurements that folds into a regular tetrahedron, and verify by making the cut-out; the same for a square pyramid.
• The cylinder's net (Part II p.82). If the two circular faces are unfolded and a cut is made along the height, you get the printed figure: two circles and a rectangle, drawn with the circles above and below the rectangle. The chapter asks what the rectangle's sidelengths are. Not in the book: one side is the height of the cylinder; the other is the circumference of the base, 2πr. That second one is the whole content of the question — the cut runs along the height, so the rectangle's other dimension has to be the distance once round the base.
• The cone's net (Part II p.82). A cone is drawn with apex O and the slant length labelled l. Slit along l and unroll. The chapter's reason, printed: every point on the boundary of the base circle is the same distance from O, so after unrolling, the net's boundary is part of a circle centred at O. A small figure of that sector-like shape is printed alongside, with O marked at its apex.
• The open cone question (Part II p.82, Math Talk). Suppose you use that same net but with O somewhere other than the centre of its curved boundary — what surface do you get? The chapter's instruction is to build the thing and see. The chapter gives no answer; it should set the model-making task.
• A third construction task (Part II p.82). Draw a net with measurements that folds into a triangular prism, and verify with a cut-out.
• The octahedron (Part II p.83). The chapter's description: glue two square pyramids together, square face to square face. A coloured drawing is captioned Octahedron. One of its nets is printed. An added reading of that net, from the printed page: eight equilateral triangles — a horizontal strip of six with one more attached above the strip and one below it, at different positions along it. The reader is asked to make the cut-out with all triangles equilateral and fold it. The chapter then states that an octahedron, like a cube, has 11 different nets.
• The dodecahedron (Part II p.83). The question is posed whether a solid can have all its faces pentagons; the answer is yes, the solid is named, a coloured drawing is captioned Dodecahedron, and the chapter states that mathematicians have determined the exact number of its nets: 43,380.
• The sphere (Part II p.84, top of page). Set as an experiment: try to make a paper cut-out that wraps a ball with no wrinkles, gaps or overlaps. Not in the book: it cannot be done, and the reason is the one that runs through the whole topic — folding moves paper without stretching it, so it preserves distances measured along the surface. A sphere's surface has no flat piece anywhere, so no flat piece of paper can match it. That is why every solid in this section is bounded by flat faces, or by surfaces like the cylinder's and cone's that are made by rolling a flat sheet up in the first place.

Figures to have open

• Fig. 4.1 (Part II p.80), redrawn, and shown step by step folding into a cube with each square colour-tracked. This is the load-bearing movement of the topic.
• The faces-edges-vertices figure (Part II p.78), redrawn for a cuboid and a triangular pyramid.
• The general prism and general pyramid, with an n-gon base drawn as a hexagon and labelled n. Standard schematic; the chapter draws only the 3-, 5- and 6-sided cases.
• All six cube-net candidates from Part II pp.80–81, redrawn to scale on a square grid, each with a fold movement. Their exact arrangements are given above; get them right, because the classification depends on the column offsets.
• The four triangle candidates from Part II p.81, redrawn. Note that the fourth has five triangles, not four.
• The cylinder net with the height and 2πr labelled, and the cone net as a sector of radius l centred at O. Both are the chapter's own figures (Part II p.82) and both need redrawing with the dimensions added — the printed cylinder net carries no dimension labels at all.
• The octahedron and its printed net (Part II p.83), and the dodecahedron (Part II p.83). Redraw.
• A ball and a flat sheet, for section 12. Standard schematic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 4, §4.2 "Visualising Solids", printed subheadings "Making Solids" (Part II pp.77–79) and "Practical Aspects of Using a Net" (Part II p.80), continuing without a further subheading to Part II p.84.
• Faces, edges and vertices, and the 6-12-8 example: Part II p.78. Prism and pyramid definitions and the two counting questions: Part II p.79. The definition of a net: Part II p.79.
• Fig. 4.1 and the practical-aspects discussion: Part II p.80. Figure it Out items 1–3: Part II pp.80–81, with item 2 carrying the Try This marker and the 11-nets statement.
• Regular tetrahedron, its four candidates, its count of 2, and two Math Talk construction tasks: Part II p.81. Cylinder and cone nets, the open Math Talk question about a non-central O, and the triangular-prism task: Part II p.82. Octahedron, its net, its count of 11, and the dodecahedron with 43,380: Part II p.83. The sphere experiment: Part II p.84.
• Part II p.102, SUMMARY, bullet 3 lists seven kinds of solid a suitable net can be folded into — the cuboid, the tetrahedron, the cylinder, the cone, the prism, the pyramid and the octahedron. The sphere is absent from that list, which is consistent with the experiment on Part II p.84.
• Forward pointer inside the same chapter: the net is used as a tool on Part II pp.84–87, which is The shortest path across a cube, found by flattening it.