Figure it Out — Nets of a Cube and a Cuboid — Exploring Some Geometric Themes

1. 13 marksGanita Prakash Cl-8 Part 2, Figure it Out, pages 80-81

   Which of the six figures shown are nets of a cube? First try to answer by visualisation, then check with cutouts.

   Hint. Count the squares first, then look for the row that becomes the band around the cube.

   The six figures are printed diagrams, so they are not reproduced here. What follows is the test to apply to each one — which is what the question is really training.

   Test 1 — count the squares. A cube has 6 faces, so a net must have exactly 6 squares. Five or seven rules a figure out immediately, with no folding needed.

   Test 2 — look for a band of four. Fold any valid net and four of its squares wrap round the cube as a band, with the remaining two closing the top and bottom. So look for a run of four squares in a line (or a run that turns a corner). Then check that of the two leftovers, one attaches to the band on one side and one on the other — two flaps on the same side of the band would both try to become the top, leaving the bottom open.

   Test 3 — the four-in-a-square rule. If any four squares meet at a single point (a 2 × 2 block), the figure is not a net. Folding a 2 × 2 block forces two of its squares onto the same face of the cube, so they overlap while another face is left bare. This one test rejects most of the wrong figures at a glance.

   Test 4 — the arrangement must be simply connected. All six squares must be joined edge to edge in one piece, with no square attached only at a corner.

   How to check by folding in your head. Choose one square as the base. Fold its neighbours up to become walls. Then see whether the remaining squares land on the four wall positions and the top, each exactly once, with none overlapping and none missing.

   Then verify physically. The book asks for cutouts for a good reason: a figure that survives all four tests should fold, and actually folding it is the proof. Rule the squares on paper at 3 cm, cut round the outside only, and crease along every internal line before folding.

   ✦ Apply four tests: exactly 6 squares; a band of four with one flap on each side; no 2 × 2 block of four squares meeting at a point; all six joined edge to edge in one piece. Any figure failing even one test is not a net, and those that survive should then be checked by cutting and folding.

2. 24 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 81

   A cube has 11 possible nets in total, where two nets are the same if one can be obtained from the other by a rotation or a flip. Find all 11 nets of a cube.

   Hint. Sort them by the longest straight run of squares — that turns a hunt into a short list.

   Organise the search instead of guessing. Classify each net by the longest straight row of squares it contains. That turns an open-ended hunt into four short cases.

   Type 1 — a row of four (6 nets). Lay four squares in a row; these become the band round the cube. The remaining two squares attach above and below the row, one on each side. Writing the four row positions as 1, 2, 3, 4, the square above can go in any of the four positions and the square below in any of the four — 16 combinations — but rotations and flips cut these down to 6 genuinely different nets. The familiar cross shape (both extra squares attached to position 2, one above and one below) is one of them.

   Type 2 — a row of three (4 nets). Three squares in a row, with the other three attached above and below in a stepped arrangement. The classic examples are the two "staircase" nets, the "T"-with-a-step, and the "S"-shaped one. There are 4 of these.

   Type 3 — a row of two (1 net). The staircase net: three pairs of squares, each pair offset from the last by one square, forming a zig-zag strip. There is exactly 1.

   Type 4 — a row of five or six. None. Five squares in a row would wrap the band and then overlap the starting square, so two faces would land on top of each other.

   The total. 6 + 4 + 1 = 11 nets.

   Two rules that stop wrong answers. · No 2 × 2 block. Any four squares meeting at one point cannot fold — two of them are forced onto the same face. · The two end squares must be on opposite sides of the band. Both on the same side and the cube is left open at the bottom.

   How to be sure you have not double-counted. Cut each candidate out, and before adding it to the list, try turning it a quarter-turn at a time and flipping it over to see whether it matches something already on the list. Working systematically by longest row makes this easy, since nets in different types can never match.

   ✦ Sorting by the longest straight row gives 6 nets with a row of four, 4 with a row of three and 1 with a row of two — 11 in all. None has a row of five or six, and none contains a 2 × 2 block of squares.

3. 33 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 81

   Draw a net of a cuboid having sidelengths (i) 5 cm, 3 cm and 1 cm (ii) 6 cm, 3 cm and 2 cm.

   Hint. A cuboid's six faces come in three matching pairs — work out the three rectangle sizes first.

   Work out the faces before drawing anything. A cuboid of dimensions l × b × h has six rectangular faces in three congruent pairs.

   (i) 5 cm × 3 cm × 1 cm

   Pair	Size	How many
   front and back	5 cm × 1 cm	2
   top and bottom	5 cm × 3 cm	2
   two ends	3 cm × 1 cm	2

   (ii) 6 cm × 3 cm × 2 cm

   Pair	Size	How many
   front and back	6 cm × 2 cm	2
   top and bottom	6 cm × 3 cm	2
   two ends	3 cm × 2 cm	2

   How to lay out the net (the cross arrangement, easiest to draw).

   1. Draw the bottom face in the middle of the page.
   2. Attach the front and back faces to its top and bottom edges — each must share the full length of the edge it is attached to, so a 5 cm edge gets a face with a 5 cm side.
   3. Attach the two end faces to its left and right edges.
   4. Attach the top face to the far edge of the front face, completing the strip of four that wraps round the cuboid.

   For (i) this gives a strip 1 + 3 + 1 + 3 = 8 cm tall and 5 cm wide, with the two end faces sticking out sideways.

   Check before you cut. · Six rectangles, in three matching pairs ✓ · Every shared edge has the same length on both faces — this is the check that catches almost all mistakes. · Total area equals the surface area: for (i), 2(5×3 + 5×1 + 3×1) = 2(15 + 5 + 3) = 46 cm²; for (ii), 2(6×3 + 6×2 + 3×2) = 2(18 + 12 + 6) = 72 cm². Add up the six rectangles in your drawing and you should get the same number.

   A note on flaps. To actually build the cuboid you would add small flaps along some edges for gluing, but flaps are not part of the net — the net is just the shape obtained by unfolding.

   ✦ (i) Two rectangles of each size 5 × 3, 5 × 1 and 3 × 1 cm, total area 46 cm². (ii) Two each of 6 × 3, 6 × 2 and 3 × 2 cm, total area 72 cm². Lay them out as a strip of four that wraps around, with the remaining two attached on opposite sides.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp204.pdf), where this is Chapter 4 (pages 70-102) — the eleventh chapter of the Class 8 course and the longest in the book. Like the rest of Part 2 it carries NO printed answer key, so every formula and count was derived and then independently recomputed in Python before being written: the Sierpinski Carpet recurrences (R_n = 8^n, H_n = (8^n - 1)/7), the Sierpinski Triangle counts (3^n and (3^n - 1)/2), the areas (8/9)^n and (3/4)^n, the Koch side count 3 x 4^n and perimeter 3 x (4/3)^n, and the face/edge/vertex formulas for prisms and pyramids (checked against Euler's relation for every case). THIS IS A HEAVILY VISUAL CHAPTER, so figure-only items are handled in one of two ways and never guessed. (1) MEASURED FROM THE RENDERED PAGE: the cube-stack count on page 97 was settled by rendering the figure at 400 dpi and observing that each row sits one step BACK as well as one step up (every bottom cube shows its full top face), which makes it a square-layered step pyramid of 16 + 9 + 4 + 1 = 30 cubes rather than the ten visible; and the three letters in the page-96 puzzle were read off the printed pixel glyphs at 700-900 dpi as C (front), A (top) and F (side). (2) FLAGGED AND ANSWERED BY METHOD: the six candidate cube nets, the projection-matching sets, the cube-combination views, the isometric figures to copy, the rolling ball and the impossible triangle are all printed diagrams; each solution gives the full method and reasoning and says plainly that the diagram is not reproduced. ONE ITEM IS LEFT OPEN BY THE BOOK ITSELF and is reported as such: the 30 x 12 x 12 shortest-path Try This on page 87, where the book computes 42 cm and 40 cm for two unfoldings (24^2 + 32^2 = 1600 verified) and then says all unfoldings must be listed to find the answer — so the solution establishes only that the shortest path is at most 40 cm. The tetracube count in the page-100 exercise was verified by exhaustive computer enumeration: 8 arrangements up to rotation, 7 up to rotation and reflection, of which 5 are flat.. Questions are referenced from the NCERT textbook for identification.