The shortest path across a cube, found by flattening it

What to assume they know

• Nets: which flat shapes fold into which solid — what a net is, and that a cuboid has several nets
• That on a flat surface the shortest route between two points is the straight segment joining them
• The Baudhayana Theorem (the relation among the squares on the sides of a right triangle), by whatever name the class has met it
• Square roots of perfect squares up to at least 1600
• Reading a labelled dimension off a drawing of a box
• The idea that a fold does not change any distance measured along the surface

What they should be able to do

• State why a path on a cuboid's surface and its image on the net have equal length
• Use that equality to argue that a particular path is shortest, rather than asserting it
• Explain why a bent image on the net proves a path is not shortest
• Recognise when a straight segment on a net fails to correspond to any path on the solid, and say what has gone wrong
• Choose a different unfolding when the first one fails, and say what changed
• Compute a path length on a net using the Baudhayana Theorem
• Compare the lengths obtained from different unfoldings of the same box and identify the shorter
• Explain why the method is not finished until all the different unfoldings have been listed

Where it usually goes wrong

• "The shortest path is obviously the one over the nearest edge." Case 4 refutes it with the chapter's own numbers: going over the top gives 42, going round gives 40. The obvious route loses.
• "You can tell by looking." You cannot, and the chapter says so before it does anything else — there are infinitely many candidates. The unfolding is what converts looking into proving.
• "Unfolding changes the distances." It does not, and this is the whole argument. Folding a sheet moves it in space but no distance measured along the sheet changes. Demonstrate with a string on a real box and the same string on the flattened box.
• "A straight line on the net is always the answer." Only if it stays on the net. Part II p.86 shows a straight segment that wanders off the paper, and such a segment describes nothing at all.
• "Any net will do." Different nets give different straight segments. That is the chapter's own moral on Part II p.86 and the reason section 12 exists.
• "40 cm is the answer." It is the shorter of two shown. The chapter's last sentence on the subsection asks for all the unfoldings to be listed, which is a statement that the problem is not yet closed.
• "The path bends when it crosses an edge, so it isn't straight." On the box it bends; on the net it does not. Straightness is a property of the flattened picture, and that is the only place the comparison is made.
• "This needs Pythagoras, which is a different chapter." The chapter names the same result the Baudhayana Theorem and uses it here without ceremony. Use the chapter's name.

Questions to check understanding

• Given a cuboid with stated dimensions and two stated surface points, find the length of a shortest surface path
• Given a path drawn on a net, decide whether it is shortest and give the reason
• Given a straight segment on a net that leaves the net, say why it is not a valid path
• Enumerate the distinct unfoldings relevant to a stated pair of points
• Apply the Baudhayana Theorem to a right triangle read off a net
• Explain in one or two sentences why unfolding does not change a path's length
• Compare two candidate routes over a box and justify which is shorter, which is the competency-based form this subsection is written in

Examples worth working on the board

Values marked not in the book are worked out here; only the 24, 32 and 40 of the third case are printed as arithmetic in the chapter.

• The set-up (Part II p.84). A hungry ant lives on the surface of a cuboid and a laddu sits on the surface too. The question stated first is the general one: on a flat surface the shortest route between two points is the straight segment, so what is the shortest route between two points on a cuboid's surface if you may travel only along the surface?
• Case 1 (Part II p.84, first drawing, marked Math Talk): the laddu is at the centre of the top face; the ant is at the centre of a side face. The positions are printed as labels on the artwork.
• Case 2 (Part II p.84, second drawing): the laddu is at the centre of an edge; the ant is again on the side face.
• The two candidate paths (Part II p.84, bottom of page, two drawings). One is drawn in red, running from the laddu across the top face and then down; the other is drawn in blue and takes a different route. The chapter asks whether either of them is shortest, and — crucially — asks first how you could ever be sure, given infinitely many possibilities.
• The hint and the flattened figures (Part II p.85, top). The hint printed is to draw the net and look at how the paths appear on it. Two nets are drawn, each carrying one of the two paths: on the first the path is a straight segment from laddu to ant; on the second it is bent.
• The chapter's own argument, in the order it makes it (Part II p.85). Carry any surface route over to the net and its length is unaltered; carry any route on the net back to the box and again nothing about its length changes. Therefore the first path is shortest, because on the net it is the straight segment between the two points, which is the shortest route there. And the second is not shortest, because on the net it is not straight. Then a third figure shows both paths on one net with the caption stating that the red path is the shortest, alongside a small box drawing with the laddu at the centre of the edge.
• The question the chapter then asks (Part II p.85, last line): has the problem of finding the shortest path between two points on a cuboid now been completely analysed? It is a rhetorical set-up for the next page.
• Case 3, the dimensioned one (Part II p.86). The cuboid is labelled 8 cm along its length, 4 cm high and 4 cm deep. The ant is at the centre of the 4 cm × 4 cm end face. The laddu sits on the bottom front edge with 2 cm marked from it to that end. The reader is asked for the shortest path.
• The failure (Part II p.86, middle figure). Unfolded in the same way as before, the net is drawn as a cross of six faces: a vertical column of four 8 cm × 4 cm faces, with a 4 cm × 4 cm end face attached to the left and another to the right of the second face down. Every edge is dimensioned — 8 cm across the top of the column, 4 cm on the short sides. The ant sits in the right-hand 4 × 4 end face; the laddu sits on the horizontal edge between the third and fourth faces of the column. Count the column faces when redrawing this: it is easy to omit the 8 × 4 face above the band that carries the two end faces, and a five-face version is not a net of an 8 × 4 × 4 box at all. The straight segment from ant to laddu is drawn and it leaves the net, crossing the empty region to the right of the third column face, so it corresponds to no path on the box at all. The chapter says so plainly and asks what to do.
• The fix (Part II p.86, lower figure, captioned as the shortest path). Unfolded a different way, the segment stays on the net and does give a path on the cuboid. The chapter's conclusion is printed as a one-line moral: the way a cuboid is unfolded matters.
• Not in the book, on case 3: the chapter prints no number for this case. Working it out is optional and probably better handed to the class, since the chapter's point here is qualitative — that an unfolding can fail — and giving a number invites students to think the number was the goal.
• Case 4, the trickier one (Part II p.87, marked Try This). The box is drawn with 30 cm along its length, and the two end faces are shown enlarged either side of it. Read off the two insets: each end face is 12 cm by 12 cm, with a 6 cm mark locating its vertical centre line. The ant is on the near end face, on that centre line, 1 cm below the top edge. The laddu is stuck to the opposite end face — the label says it is stuck to the back of the box — on its centre line, 1 cm above the bottom edge. So the box is 30 cm × 12 cm × 12 cm.
• The first unfolding (Part II p.87, left figure). A T-shaped net: a horizontal bar of three faces with the laddu in the left face, the long face in the middle and the ant in the right face, and a vertical strip of three more faces hanging below the centre. The straight segment from laddu to ant is horizontal and is labelled 42 cm.
• The second unfolding (Part II p.87, right figure). A vertical strip of four long faces, with the ant's end face attached to the right of the topmost and the laddu's end face attached to the left of the third one down. The straight segment between them is the hypotenuse of a right triangle whose legs are labelled 24 cm and 32 cm.
• The chapter's arithmetic, exactly as printed (Part II p.87). Because there is a right triangle, the Baudhayana Theorem applies: d² = 24² + 32², so d = √1600 = 40 cm. This is the only computed length in the whole subsection.
• Not in the book, where the printed numbers come from — and this is the payoff of section 11. In the first unfolding the 42 is 1 + 30 + 11: one centimetre from the ant up to the top edge of its face, thirty across the long top face, and eleven down the far end face to the laddu, which sits 1 cm above the bottom of a 12 cm face. In the second unfolding the box is opened out around its length, so the four long faces lie in a strip and the two end faces hang off it: 32 is 1 + 30 + 1, because each end face is now entered across the edge the ant and the laddu are each 1 cm from; and 24 is 6 + 12 + 6: half the width of the long face the ant's end face hangs from, the full 12 cm of the next long face, and half the width of the long face the laddu's end face hangs from — both sit on centre lines, 6 cm from the side edges, as the printed figure marks. Then 24² + 32² = 576 + 1024 = 1600 and d = 40. Both printed numbers reconstruct exactly from the printed dimensions, which is the strongest available check that the 12 × 12 end faces have been read correctly.
• What the chapter concludes (Part II p.87). The line-segment lengths differ from one unfolding to the next, so the possible unfoldings must be listed carefully to find the answer. The chapter stops there. It does not claim that 40 cm is the minimum, only that the two unfoldings shown give different lengths. An explanation must not upgrade that.
• Not in the book, an honest statement for the class: of the two unfoldings the chapter draws, 40 cm is shorter than 42 cm, so the ant does better going round the box than over the top. Whether some third unfolding beats 40 is exactly the open question the chapter hands over.

Figures to have open

• A real box and a piece of string, or a convincing movement of one, for section 3. This is the only place the length-preserving claim can be made to feel true rather than asserted, and the chapter has no figure for it.
• The two cases from Part II p.84, redrawn: laddu on the top face and laddu on an edge, with the ant on the side face and both candidate paths in two colours.
• The two flattened nets from Part II p.85 carrying those paths, plus the combined net showing both. Redraw; keep the colour coding, since the caption's verdict depends on it.
• The 8 cm × 4 cm × 4 cm box and both of its unfoldings from Part II p.86, fully dimensioned, with the off-net portion of the failed segment clearly outside the paper. Redraw — this is the topic's most important single figure and the excursion is easy to miss.
• The 30 cm × 12 cm × 12 cm box from Part II p.87 with both end faces enlarged and the 6 cm centre marks and 1 cm offsets labelled. Redraw.
• Both unfoldings from Part II p.87, with 42 on the first and the 24–32–40 right triangle on the second. Redraw, and label the pieces that make up 42, 32 and 24 — the printed figures give the totals only.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 4, §4.2 "Visualising Solids", printed subheading "Shortest Paths on a Cube", Part II pp.84–87. The subsection ends where "Representation of Solids on a Plane Surface" begins, part-way down Part II p.87.
• The set-up and the first two cases with their candidate paths: Part II p.84, carrying a Math Talk marker. The hint, the flattened paths and the chapter's length-preserving argument: Part II p.85. The dimensioned 8-4-4 case, the failed unfolding and the working one: Part II p.86. The 30-12-12 case, the two unfoldings, the Baudhayana step and the closing moral: Part II p.87, carrying a Try This marker.
• Part II p.102, SUMMARY, bullet 4 states that the shortest path between two points on a cuboid's surface can be found using a suitable net — note the word suitable, which carries the whole content of Part II p.86.
• Backward pointer inside the same chapter: nets themselves are established on Part II pp.79–84, which is Nets: which flat shapes fold into which solid.
• The subsection heading names a cube; every worked case is a cuboid. The chapter's own text on Part II p.84 asks the question about a cuboid.