NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –81Nets of a cube — Figure it Out

Book page 80 Updated on2026-09-05

Q1.
Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try.
Answer

Writing each figure on squared paper, with a dot for an empty cell:

FigureArrangement of the six squaresNet of a cube?
(i)Two squares on top, four in a row below, the row starting under the second squareNo
(ii)A row of three, and a second row of three below it overlapping in one columnYes
(iii)A staircase — three rows of two, each row stepped one place to the rightYes
(iv)A row of four with one square above and one square below the third columnYes
(v)A row of four with two squares hanging one below the other from the second columnNo
(vi)A row of four with one square above the third column and one below the secondYes

So (ii), (iii), (iv) and (vi) fold into a cube; (i) and (v) do not.

Why (iv) and (vi) work. They are of the ‘1 – 4 – 1’ kind. Roll the row of four into a band — that band is the four side walls. One extra square is above the row and one below, so one closes the top and the other the bottom. It does not matter which column each extra square hangs from.

Why (v) fails. Its two extra squares are both below the row, stacked in the same column. Roll the row of four into a band; the first extra square becomes the bottom face. The second is attached to the far edge of that bottom face, and that edge already belongs to the wall opposite the one it came from. So the sixth square lands on a wall that is already there, and the top of the cube is left open.

Why (i) fails. Same trouble. Both extra squares are on one side of the row of four. Fold the row into a band and let the square directly above the row become the lid. The last square is joined to that lid along an edge which the lid already shares with one of the four walls — so it folds down on top of a wall, and the base is never covered.

The quick test: find a row of four. It must become the band of side walls, so the remaining two squares must lie one on each side of that row — one to close the top and one to close the bottom. If both are on the same side, you get an overlap and an open face. That single check settles (i), (iv), (v) and (vi) at a glance.
Q2.
A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be obtained from the other by a rotation or a flip. For example, the following nets are all considered the same — Find all the 11 nets of a cube.
Answer

The tidiest way to hunt them down is by the length of the longest straight row. Below, X marks a square and a dot marks an empty cell.

Type 1 – 4 – 1 (six nets). A row of four, with one square above and one below. The one below can sit under any of the four columns, and the one above under any of the four — but rotations and flips cut the sixteen possibilities down to six.

1X . . .
X X X X
. . . X
2X . . .
X X X X
. . X .
3X . . .
X X X X
. X . .
4. X . .
X X X X
. . X .
5. X . .
X X X X
. . . X
6. X . .
X X X X
. X . .

Type 2 – 3 – 1 (three nets). The longest row has three squares; two squares sit above it and one below (or the mirror image).

7X X . .
. X X X
. X . .
8X X . .
. X X X
. . X .
9X X . .
. X X X
. . . X

Type 3 – 3 (one net). Two rows of three, overlapping in a single column — this is figure (ii) of Question 1.

X X X . .
. . X X X

Type 2 – 2 – 2 (one net). The staircase — figure (iii) of Question 1.

X X . .
. X X .
. . X X
6 + 3 + 1 + 1 = 11 nets
Why there is no 5 – 1 or 6 net: a straight row of five would have to wrap round a cube of only four walls, so two of its squares would land on the same face. And a straight row of six is worse still. That is why the longest row in any cube net is four.
Q3.
Draw a net of a cuboid having sidelengths: (i) 5 cm, 3 cm, and 1 cm (ii) 6 cm, 3 cm, and 2 cm
Answer

Use the same ‘1 – 4 – 1’ layout as the cube net, but now the six rectangles come in three matching pairs.

(i) 5 cm × 3 cm × 1 cm. Lay a horizontal strip of four rectangles that wraps round the 5 cm length, then attach the two 3 cm × 1 cm ends.

5 × 3 5 × 3 3×1 3×1 5 × 1 5 × 1
Net of a 5 cm × 3 cm × 1 cm cuboid: two 5 × 3 faces, two 5 × 1 faces and two 3 × 1 faces.

(ii) 6 cm × 3 cm × 2 cm. Exactly the same layout with the numbers changed: two rectangles 6 × 3, two 6 × 2 and two 3 × 2.

Surface area (i) = 2(5×3 + 5×1 + 3×1) = 2(15 + 5 + 3) = 46 cm²
Surface area (ii) = 2(6×3 + 6×2 + 3×2) = 2(18 + 12 + 6) = 72 cm²
Check it yourself: in any correct net of a cuboid, edges that will be glued together must be equal in length. Run your finger round the boundary of your drawing and pair up the free edges — every pair should match.
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