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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –85Shortest Paths on a Cube — In-text Questions

Book page 84 Updated on2026-09-05

Q1.
Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.
Answer

It cannot be done. A sphere has no net.

A cylinder and a cone flatten out because they are curved in only one direction — every point of them lies on a straight line drawn on the surface. A sphere carries no straight lines at all; it curves in every direction at once. So paper, which will bend but not stretch, can never lie flat against it.

Why paper refuses: take the two ends of a strip of paper and try to press them onto a ball. Near the middle the paper touches, but at the edges it must either be stretched (it tears) or gathered up (it wrinkles). Distances on a sphere simply do not match distances on a plane: on a plane, a circle of radius r has circumference 2πr; on a sphere, a circle drawn at distance r from a point has circumference less than 2πr. Something has to give.
Did you know? This is why every flat map of the Earth is wrong somewhere. Greenland looks as big as Africa on many wall maps, though Africa is about 14 times larger. It is also why a football is not made from one piece of leather but from many small panels, and why an orange peel will not lie flat however carefully you press it.
Q2.
What is the shortest path for the ant to reach the laddu?
Answer

The ant should walk in a straight line on the net — that is, straight up the side face and then straight across the top face, the two straight bits meeting the top edge at the same point.

The idea. Unfold the box so that the side face carrying the ant and the top face carrying the laddu lie flat next to each other. On this flat figure, join the ant to the laddu by a straight line. Now fold the box back up: the line bends over the edge and becomes a path on the surface — and its length has not changed.

On the surface: a bent path, hard to compare
On the net: a straight segment — and a straight segment is the shortest path between two points on a plane
Why this proves it is shortest: every path on the box turns into a path of exactly the same length on the net, and every path on the net turns back into a path of the same length on the box. So the two problems have exactly the same set of lengths. On the flat net we already know the winner — the straight line. That is why the answer must be the path that straightens out.
Q3.
What about in the following case?
Answer

The same method works when the laddu sits at the middle of an edge — but now the laddu is on the boundary between two faces, so two different unfoldings are open to the ant.

  • Go across the side face and then over the top face to the edge.
  • Go across the side face and then over the front face to the same edge.

Unfold each way, draw the straight line, measure both, and take the shorter. When the laddu sits exactly at the midpoint of the edge and the ant at the centre of the face, the two come out equal — but in general they will not, and the ant must compare.

Tip: Never settle for the first unfolding that looks reasonable. A point on an edge belongs to two faces, and a point at a corner belongs to three.
Q4.
If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities?
Answer

By turning the question into one we can already answer. Unfold the box; the path becomes a path on a flat net, with its length unchanged. On a plane we know that the straight line is shortest — so if our path unfolds into a straight line, nothing can beat it.

path on cuboid ⟷ path of the same length on the net
shortest path on the net = straight line
so shortest path on the cuboid = the one that unfolds straight

Two things make this a proof rather than a guess:

  • The correspondence works both ways. Any rival path on the box also becomes a path on the net, and it cannot be shorter than the straight line there.
  • Lengths are exactly preserved. Unfolding bends the surface but never stretches it, so no length is lost or gained.
Why we cannot simply try lots of paths: there are infinitely many, so testing is hopeless. Mathematics gets round this by changing the setting to one where the answer is already known — that is the real lesson of this section.
Q5.
For example, are either of these the shortest path?
Answer

Only the first one is. Draw the net and see what each path looks like on it.

PathOn the net it becomesShortest?
The first (red) pathA straight segment from ant to ladduYes
The second pathA bent line — it changes direction at the edgeNo

The test is simple: unfold, and look at whether the path comes out straight. A path that still has a kink in it on the flat net can always be shortened by pulling it taut.

Q6.
What does this show?
Answer

It shows that the problem of the shortest path on a cuboid is the same problem as the shortest path on its net.

Every path on the surface → a path of equal length on the net
Every path on the net → a path of equal length on the surface

Because the two collections of paths match up length for length, the shortest in one is the shortest in the other. That is why the first path in the picture is shortest — it becomes the straight line joining the ant and the laddu — while the second is not, because it becomes a bent line.

Why lengths survive the unfolding: unfolding only turns faces about the edges they share. Nothing is stretched, squeezed or torn, so a length measured along the surface is the same length after it is laid flat. This is the same fact that let us find the curved surface area of a cone by flattening it into a sector.
Q7.
Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?
Answer

No, not yet — and the next two pages show exactly what is missing.

We have proved that the shortest path must be a straight line on some net. But a cuboid can be unfolded in many different ways, and:

  • on some nets the straight segment between the two points runs outside the net, so it does not correspond to any path on the box at all;
  • different nets give straight segments of different lengths.
Same box, same two points, three unfoldings:
one gives 42 cm, another gives 40 cm, a third gives something else

So the method is only finished when we list all the sensible unfoldings, work out each straight-line length, and take the smallest. That is what the worked examples on pages 86 and 87 do.

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