Nets of solids
How do you make a box, a tent or a tin from one flat sheet, and how can you tell before cutting whether the pattern will close up?
Where you are 8 in Solids
When you reach for it
A cereal box, a tent and a tin all start as flat sheets. Before cutting, you want to know what shape to cut so that, folded up, it closes into the solid with every face covered once. That flat pattern is a net. You reach for one when you design a package or a model, when you want to see every face of a solid at once, and before adding up the areas of the faces in Surface area, and how it compares with volume.
This lesson takes two days. Part 1 (Examples 1 to 4) is nets of prisms: what a net is, which patterns fold, and which lengths must match. Part 2 (Examples 5 to 7) is nets of pyramids and cylinders. The solids and their faces, edges and vertices are named in Prisms and pyramids, Polyhedra and Cylinders and cones.
The procedure
Part 1: nets of prisms
What a net is. A net of a solid is a flat shape in one piece, made of the solid’s faces joined edge to edge, that folds along those joins into the surface of the solid with every face covered exactly once, no gap and no overlap. The joins are the folds, dashed in these figures. Folding bends the sheet at a fold and nowhere else, so a net has exactly one piece for each face, each the same shape and size as its face.
You can also get a net by cutting a cardboard box along some of its edges until it lies flat in one piece; the uncut edges become the folds. Different cuts give different nets. A cube has eleven, counting a net and its mirror image or a turned copy as one.
Reading a row of squares. In Figure 1 the four squares of the row fold up one after another and wrap round the four sides of the cube. Two squares with one square between them end up opposite each other: each fold is a quarter turn, and two quarter turns make a half turn. The squares above and below the row close the top and the bottom.
How a pattern fails. A pattern can have the right pieces and still not be a net.
- A face is missing, and the folded shape has a hole.
- Two pieces land on the same face and overlap, so some other face is left open.
- Edges that meet on folding have different lengths, leaving a gap or an overhang.
A piece attached in the wrong place does one of the last two things.
Edges that must match. Every edge of the solid is either a fold, where the two faces were never separated, or a cut, and then it appears twice on the outline of the net, once on each face it borders. Folding brings those two outline edges together. So the outline edges come in matching pairs of equal length, and an unlabeled length is found from the edge it will meet.
The net of a prism. A right prism (side faces standing straight up) has two bases and one rectangle for each side of a base. The usual net puts the rectangles side by side in a strip as tall as the prism, with a base attached to the top of the strip and the other to the bottom. Each rectangle wraps one side of the base, so its width equals that side, and the strip is as long as the perimeter of the base. For a box the rectangles alternate between two widths, because opposite sides of a rectangle are equal.
Part 2: nets of pyramids and cylinders
The net of a pyramid. A pyramid has one base and one triangle for each side of the base. A tidy net puts the base in the middle with a triangle on each side, like a star. Each triangle’s bottom side equals the side of the base it is attached to, and when the triangles fold up, each slanted side meets a slanted side of the next triangle round, so those two must be equal.
The triangles must also be tall enough. Take a square base of side 10 cm and four identical triangles, each with equal slanted sides, so the tips can meet only above the center, 5 cm in from every side. A tip gets furthest in when its triangle is folded flat onto the base, and then it is exactly as far in as it stuck out from the base side in the net. So the tips can meet only if each sticks out more than 5 cm, half the side of the square; at exactly 5 cm they meet only lying flat. With more, they do meet: standing straight up a tip is 0 cm in, folded flat it is past the center, so on the way it passes 5 cm in, and the four, being alike, get there together. That is the rule for a square base; other bases need other numbers.
The net of a cylinder. A cylinder’s curved side unrolls like the label peeled off a tin: cut straight up and laid flat, it is a rectangle. One pair of its sides is the height. The other pair wrapped once around the circle, so each is as long as the distance around the circle, its Circumference; for a tin wider round than it is tall, those are the long sides. The net is that rectangle with one circle touching each of the two sides that wrapped around, one to close each end of the tube.
A cone’s net is a circle and a fan-shaped piece of a larger circle, a sector, whose size is worked out in Sectors, segments and their areas.
Counting folds. The folds of a net never form a loop, a ring of pieces joined fold to fold all the way round: such a ring wraps all the way round part of the solid, like a bracelet, and a bracelet cannot lie flat in one layer until it is cut. So every net can be drawn by starting with one piece and attaching each new piece along one fold to a piece already drawn. Each new piece brings one fold, so a net of P pieces has P - 1 folds. Every other edge was cut and shows up as a matching pair, so
Worked examples
Which six squares fold into a cube
Cassia wants to cut the net of a toy cube block from card. Which of the patterns P, Q, R and S in Figure 2 fold into a cube?
Each pattern has six squares, so the question is whether two land on the same face.
In P the row of four wraps round the four sides, as in Figure 1, and the squares above the second and below the fourth close the two ends. Six different faces.
In Q the four squares on the left meet at one point, which would become a corner of the cube with four faces round it. Only three faces of a cube meet at a corner, so two squares overlap.
In R, take the first square as the bottom. The second stands up as the right wall and the third folds over as the top. The fourth hangs down from the top as the back wall, and the last two swing round from it as the left and front walls. Six different faces.
In S the row wraps round, making its first and fourth squares neighbors, and the squares above them both fold over the same end. They overlap and the other end stays open.
So P and R fold into a cube; Q and S do not.
Finding lengths on a block’s net
Davor draws the net of a toy building block shaped like a box (Figure 3) and labels only three lengths: the first rectangle of the strip is 8 cm wide and 2 cm tall, and the lid above it is 3 cm deep. Find the width of every rectangle in the strip, the size of each lid and the length of the strip.
The lid hangs from the 8 cm top edge of the first rectangle, so it is 8 cm by 3 cm. Folded, the lid’s 3 cm right side comes down onto the top of the second rectangle, so that rectangle is 3 cm wide, and the lid’s far 8 cm edge lands on the third, so it is 8 cm wide. The fourth is opposite the second, 3 cm, and its free side meets the free side of the first, both 2 cm. The bottom lid covers the face opposite the top lid, so it is also 8 cm by 3 cm. The strip is
long and 2 cm tall. Check: 22 cm is the distance round the lid, as it should be, since the strip wraps once round the block.
A dog kennel with a sloping roof
Ebba plans a dog kennel with a sloping roof and first makes it from card (Figure 4). Each end is a trapezoid with floor 48 cm, back wall 40 cm, front wall 54 cm and roof 50 cm along the slope, and the kennel is 70 cm long; the door is cut later. List the pieces of the net, find the length of the strip, and say what goes wrong if the roof and the front wall swap places.
The kennel is a prism on the trapezoid ends. Its net has the two trapezoids and four rectangles 70 cm long: floor 48 cm wide, front wall 54 cm, roof 50 cm and back wall 40 cm, in the order met going round the trapezoid. The strip is 70 cm tall and
long, the perimeter of the trapezoid.
Each end hangs from the floor by its 48 cm side. As the walls fold up, its 54 cm side meets the end of the front wall, its 50 cm slope the end of the roof, and its 40 cm side the end of the back wall: every pair matches. Counting, 6 pieces give 5 folds, and the prism has 12 edges, so there are 12 - 5 = 7 matching pairs: three round each end, and the two free 70 cm sides of the strip.
If the roof and front wall swap, the 50 cm roof folds up beside the floor and its end meets the trapezoid’s 54 cm side. The lengths do not match, so the pattern is not a net of this kennel.
Where the second end can go
Gaspar’s wedge block is a triangular prism 10 cm long whose ends are triangles with sides 5, 5 and 6 cm. His net so far (Figure 5) is a strip of three rectangles 5, 6 and 5 cm wide, with one triangle on top of the middle rectangle, attached by its 6 cm side. Where can the second triangle go?
The first triangle closes the end at the top of the strip, so the second must close the end at the bottom; on any top edge it would overlap the first. It must also hang by a side as long as the edge: by its 6 cm side under the middle rectangle, or by a 5 cm side under an outer rectangle, turned so that its 6 cm side lies next to the middle rectangle, whose 6 cm bottom edge it meets on folding. The free 10 cm sides of the strip are out, since no side of the triangle is 10 cm; they meet each other.
So there are three places, one under each rectangle.
The net of a pyramid block
Hedvig’s toy pyramid block has a square base of side 6 cm, and each triangle has a 6 cm bottom side and two 5 cm slanted sides (Figure 6). On the net each tip sticks out 4 cm from its base side. Which edges fold, which meet, how many edges does the block have, and do the tips meet?
The 5 pieces have 4 folds, the sides of the square, where each triangle’s 6 cm side matches a side of the base. The outline is the eight slanted sides. Folding up, each triangle’s right slanted side meets the left slanted side of the next one, 5 cm against 5 cm: 4 matching pairs.
The block has 4 + 4 = 8 edges, four of 6 cm round the base and four of 5 cm up to the apex. The tips meet because each sticks out more than half the base side: 4 > 3.
A cylinder block, rolled two ways
Cassia’s cylinder block is 7 cm tall, and a strip of paper wrapped once around it measures 22 cm. Describe its net. If the same rectangle were rolled the other way, what tube would it make, and would the block’s circles close it?
The curved side unrolls into a rectangle 22 cm by 7 cm. The 22 cm sides wrapped around, so each circle is 22 cm round, and the 7 cm sides are the height. One circle touches each 22 cm side (Figure 7).
Rolled the other way, the 7 cm sides wrap round, making a tube 22 cm tall and only 7 cm round. Its ends need circles 7 cm round, so the block’s circles do not fit. The circles decide which way the rectangle rolls.
Counting folds and matching edges
A net of one polygon and ten triangles folds into a pyramid. What is the base? How many edges, folds and matching pairs are there?
SolutionA pyramid has one triangle per side of its base, so the base has 10 sides, a decagon. The pyramid has 10 edges round the base and 10 up to the apex, 20 in all. The net has 11 pieces, so 11 - 1 = 10 folds and 20 - 10 = 10 matching pairs.
Count again from the outline. The pieces have 10 + 10 \times 3 = 40 sides, and each fold uses up two of them, leaving 40 - 2 \times 10 = 20 outline sides, which make 10 pairs, as before.
Boundaries
Every solid in this lesson has many nets made only of polygons. Whether every solid with flat faces that bulges outward has a net that does not overlap itself is still an unsolved question. A cylinder or cone needs curved pieces, and a sphere has no net at all: no flat piece bends onto a sphere without stretching or wrinkling, which is why every flat map of the whole Earth distorts something.
The prisms here are right prisms. An oblique prism’s side faces are parallelograms, so its strip is made of parallelograms.
A net joins pieces along whole edges and folds only there. The glue tabs on a real card model are not part of the net.
Where this goes next
A net lays every face out flat, which is what adding up face areas needs: Surface area, and how it compares with volume does that, Surface area of prisms and cylinders turns the strip and two bases into a formula, and Surface area of pyramids and cones does the same for the star of triangles. The cone’s fan returns in Sectors, segments and their areas, and a net drawn smaller than the real object uses the scales of Using a scale on maps, plans and models.
Exercises
Part 1: does it fold into a cube?
- Does pattern A in Figure 8 fold into a cube? Does pattern B? If a pattern fails, say what goes wrong.
- Answer the same questions for C and D.
- Answer the same questions for E and F.
- A row of four squares, numbered 1, 2, 3, 4 from left to right, has square 5 attached above square 2 and square 6 below square 4. It folds into a cube. Which pairs of squares end up opposite each other?
- Gaspar numbers a cube block like a die, so opposite faces should add to 7. His net is a row of four squares numbered 1, 2, 6, 5, with a 3 above the 2 and a 4 below the 6. Do all opposite faces add to 7?
Part 1: lengths on prism nets
- A box net has a strip of four rectangles, each 5 cm tall, with widths 12, 7, 12 and 7 cm, and a lid above and below the first rectangle. What size is each lid, how long is the strip, and what are the box’s dimensions?
- The strip in a box’s net is 30 cm long and its first rectangle is 9 cm wide. How wide are the four rectangles?
- A wedge block is a triangular prism 9 cm long whose ends are right triangles with sides 3, 4 and 5 cm. Describe the strip in its net.
- A larger kennel has trapezoid ends with floor 60 cm, back wall 45 cm, front wall 70 cm and roof 65 cm, and is 80 cm long. Ebba builds a model with every length one fifth of the real one. How wide is each rectangle in the model’s strip (floor, front, roof, back), how tall is the strip, and how long?
- A tea shop sells tea in a box shaped like a hexagonal prism: each end is a hexagon with six 4 cm sides, and the box is 15 cm tall. How many pieces does its net have, what are the rectangles, and how long is the strip?
- A pattern for a box has a strip of four rectangles 6 cm tall with widths 10, 4, 10 and 4 cm, and lids 10 cm by 5 cm above and below the first rectangle. Explain why it does not fold into a box.
Part 1: reasoning
- Explain why no net of a cube contains four squares in a 2 by 2 block.
- Explain why the outline of every cube net is made of exactly 14 square sides.
Part 2: pyramids
- A net of a hexagonal pyramid has a hexagon in the middle. How many pieces, folds and matching pairs does it have?
- A square pyramid has base sides of 9 cm and slanted edges of 11 cm. In its star-shaped net, how many outline edges are there, and how long is the whole outline?
- Hedvig cuts out an equilateral triangle with 16 cm sides and folds along the three lines joining the midpoints of its sides. What solid does she get, and how long is each edge?
- A square pyramid net has a base of side 14 cm and four identical triangles with equal slanted sides. In one version each tip sticks out 6 cm from its base side; in another, 9 cm. Which version folds into a pyramid?
Part 2: cylinders and naming
- A paper label goes once round a tea tin, with 2 cm extra overlapping for glue. The label is 32 cm long and 11 cm tall. What rectangle is in the net of the tin, and how far round is each circle?
- A sheet 18 cm by 40 cm is rolled into a tube with no overlap. Describe the two tubes it can make.
- A tea tin is 14 cm tall and 26 cm round. Davor draws its net at half size. What size is the rectangle in his drawing, and how far round is each circle?
- Name the solid each net folds into: (a) two pentagons and five rectangles; (b) a hexagon and six triangles; (c) two circles and a rectangle; (d) four equilateral triangles.
Part 2: reasoning
- A prism has 9-sided bases. Find the number of pieces, folds and matching pairs in its net, counting the pairs from the outline, and check that folds plus pairs gives the prism’s number of edges.
- Explain why the two circles in a cylinder’s net cannot both touch the same side of the rectangle.
- Explain why, in a net of any solid with flat faces, the folds and the matching pairs together number exactly the edges of the solid.
Answers
- A folds into a cube. B does not: a row of four already goes all the way round, so the fifth square lands on the first.
- C folds into a cube. D does not: its 2 by 2 block would put four faces round one corner.
- E folds into a cube. F does not: with the left square as the bottom, the third is the top, the column runs down the back, and its last square comes round onto the bottom, on top of the first.
- 1 and 3, 2 and 4 (one square between them in the row), and 5 and 6 (one above the row, one below).
- Yes. 1 and 6 are opposite, and so are 2 and 5; 3 and 4, one above the row and one below, close opposite ends. Each pair adds to 7.
- Each lid is 12 cm by 7 cm. The strip is 12 + 7 + 12 + 7 = 38 cm long and 5 cm tall, and the box is 12 cm by 7 cm by 5 cm.
- 9, 6, 9 and 6 cm: the widths alternate, so 30 - 9 - 9 = 12 cm is shared by two equal rectangles.
- Three rectangles 3 cm, 4 cm and 5 cm wide, each 9 cm tall; the strip is 3 + 4 + 5 = 12 cm long and 9 cm tall.
- 12, 14, 13 and 9 cm wide, each 16 cm tall (every length divided by 5). The strip is 12 + 14 + 13 + 9 = 48 cm long.
- 8 pieces: two hexagons and six rectangles, each 4 cm by 15 cm. The strip is 6 \times 4 = 24 cm long.
- The lids’ 5 cm sides fold down onto the top edges of the 4 cm rectangles. A 5 cm edge cannot meet a 4 cm edge; the lids should be 10 cm by 4 cm.
- Four squares in a 2 by 2 block share one corner point, so they would put four faces round one corner of the cube. Only three faces meet at a corner, so two would overlap.
- Six squares have 6 \times 4 = 24 sides. The 5 folds each hide two sides, one from each square they join, leaving 24 - 10 = 14 on the outline.
- 7 pieces (a hexagon and six triangles), 6 folds and 6 matching pairs; 6 + 6 = 12 edges.
- There are 8 outline edges, two slanted sides on each of four triangles; the outline is 8 \times 11 = 88 cm.
- A triangular pyramid with four equilateral faces. The folds cut each side in half, so every edge is 16 \div 2 = 8 cm.
- A tip must stick out more than 14 \div 2 = 7 cm, so only the 9 cm version folds into a pyramid.
- The part that wraps exactly once is 32 - 2 = 30 cm, so the rectangle is 30 cm by 11 cm and each circle is 30 cm round.
- Either a tube 18 cm tall and 40 cm round (the 40 cm sides wrap) or a tube 40 cm tall and 18 cm round (the 18 cm sides wrap).
- Halving every length, the rectangle is 13 cm by 7 cm and each circle is 13 cm round.
- (a) A pentagonal prism. (b) A hexagonal pyramid. (c) A cylinder. (d) A triangular pyramid.
- 11 pieces, so 10 folds. The pieces have 2 \times 9 + 9 \times 4 = 54 sides; the folds use 20, leaving 54 - 20 = 34 on the outline, which is 17 matching pairs. Check: 10 + 17 = 27 = 3 \times 9 edges.
- Each side that wraps round becomes the rim at one end of the tube. Two circles on the same side would both close that end and leave the other open.
- Each edge is either left joined, a fold, or cut, giving two outline sides that come back together, a matching pair. So every edge is counted once.