NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –87Shortest Paths on a Cube — In-text Questions
Book page 86 Updated on2026-09-05
Q1.
Find the shortest path between the ant and the laddu in the following case:
Answer
The box is 8 cm × 4 cm × 4 cm. The ant is at the centre of the 4 cm × 4 cm end face, and the laddu is on the bottom front edge, 2 cm from that end.
Unfold the end face and the front face into one plane. Measure from the vertical edge where they meet, taking that edge as the zero line.
On the end face: the ant is 2 cm from the edge (half of 4) and 2 cm up On the front face: the laddu is 2 cm from the same edge, on the floor line Horizontal separation = 2 + 2 = 4 cm Vertical separation = 2 − 0 = 2 cm
d² = 4² + 2² = 16 + 4 = 20 d = √20 = 2√5 ≈ 4.47 cm
The two faces unfolded flat. The straight segment is the ant’s shortest route, and the Baudhayana Theorem gives its length.
Why the book’s first attempt failed. In the cross-shaped net drawn first, the straight segment joining the ant to the laddu passes outside the net, over empty paper. A line that leaves the net does not correspond to any walk on the box, so that unfolding is useless here. Re-unfolding the box so that the ant’s face is hinged onto the face the laddu is on puts the whole segment inside the net, and then it is a genuine path.
Check it yourself: the laddu sits on the edge shared by the front face and the bottom face, so the ant could equally well go across the bottom. Unfold that way and you get 2 cm and 4 cm again — the same 2√5 cm. Both routes tie.
Why the way you unfold matters: unfolding is a choice, and each choice tests one family of routes. A route is only found if some unfolding lays out exactly the faces it crosses, side by side, with the whole segment inside them.
Q2.
So what do we do now?
Answer
Unfold the cuboid a different way — one that lays out the faces the ant would actually walk over, so that the straight segment stays inside the net.
Decide which faces the path could cross.
Unfold so that exactly those faces lie flat, side by side, in the order the ant would meet them.
Join the two points with a straight line and check that it stays inside the net.
Measure it with the Baudhayana Theorem.
Repeat for every other reasonable set of faces, and keep the smallest length.
Why a segment leaving the net is meaningless: the net is the box’s surface, opened up. Paper outside the net is not part of the box. A line crossing it does not correspond to any walk the ant could take, so its length tells us nothing.
Q3.
What is the length of the shortest path between the ant and the laddu?
Answer
The box is 30 cm long with 12 cm × 12 cm ends. The laddu is stuck on the back end face, on its centre line, 1 cm above the bottom; the ant is on the front end face, on its centre line, 1 cm below the top.
The shortest path is 40 cm.
Unfold so that the ant goes down over the bottom, round one side, over the top, and on to the far face. Lay out the four long faces as one strip: bottom, side, top — that is 12 + 12 + 12 = 36 cm across the strip. Hinge the laddu’s face onto the bottom and the ant’s face onto the top.
Along the length of the box: 1 cm (down the back face to the bottom edge) + 30 cm (the box) + 1 cm (up the front face to the top edge) = 32 cm
Across the strip: the laddu is 6 cm along the bottom edge, the ant is 6 cm along the top edge 6 (across the bottom) + 12 (across the side) + 6 (across the top) = 24 cm
d² = 32² + 24² = 1024 + 576 = 1600 d = √1600 = 40 cm
Compare the unfoldings. The book shows two of them; there are others, and they are all worth testing.
Route
Legs of the right triangle
Length
Back face → bottom → top → front face (round one side)
32 cm and 24 cm
40 cm — shortest
Back face → bottom → a side → front face
37 cm and 17 cm
√1658 ≈ 40.7 cm
Back face → bottom → front face (straight along)
42 cm and 0
42 cm
Back face → one side → front face
42 cm and 10 cm
√1864 ≈ 43.2 cm
Why the winning route looks so roundabout: going straight along the bottom seems obvious, but it makes the ant climb 1 cm down at one end and 11 cm up at the other — the two ends fight each other. Curling round the box lets the 1 cm at each end be used in the same direction, and the extra 24 cm travelled sideways buys a much shorter straight line. That is the surprise of this problem: on a box, the natural-looking route is often not the shortest, and only a careful list of unfoldings finds the winner.
Check it yourself: 24, 32, 40 is the 3, 4, 5 triangle multiplied by 8 — a Baudhayana triple, so the answer comes out a whole number.