NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –97Views of solids built from cubes — Figure it Out

The method. Write the solid down as a plan with heights — a small grid, one cell for each square of floor the solid stands on, with the number of cubes stacked there written in. Then each view is read off mechanically:

• Top view — the plan itself: shade every cell where the height is at least 1.
• Front view — for each column of the plan (left to right), draw a bar as tall as the largest height in that column.
• Side view — for each row of the plan (front to back), draw a bar as tall as the largest height in that row.

The word largest is the whole trick: a view is a silhouette, so a tall stack hides everything shorter behind it.

Solid (i) — 3 cubes, all one layer high: two side by side at the front and one behind the right-hand cube.

Solid (ii) — 4 cubes: a one-cube-wide arm running two cubes back, with a two-cube column standing beside the back cube of the arm.

Solid (iii) — 4 cubes, all one layer high: a row of three at the back with one more sticking out in front of the left-hand end.

Solid (iv) — 6 cubes: a 2 × 2 block of floor, the back row stacked 2 high and the front row 1 high (a two-step staircase).

Solid (v) — 4 cubes, only one cube deep: a column of three with one cube joined to the middle one on the right.

Solid (vi) — the tallest of the six: a one-cube-high base with a notch cut into its front edge, a column rising two cubes above the base towards the back, and one further cube standing on the base at the right. Write its plan with heights straight off the picture in your book and then apply the three rules above; because the base has a notch, the top view has a bite taken out of it, and because the column rises two cubes, the front view is a stepped outline three cubes tall.

The C is 3 cubes wide, 4 cubes tall and just 1 cube deep: a row of 3 at the top, a row of 3 at the bottom, and one cube in the left-hand column on each of the two middle levels — 3 + 1 + 1 + 3 = 8 cubes.

(i) Because the whole letter is only one cube deep:

(ii) Making the top view an H. The top view is the plan of the solid, so we must add cubes behind the C to fill out an H-shaped floor plan. The trick is that the front view will not change as long as every new cube is placed at a height where that column of the C is already filled.

(iii) Making the side view an F as well. Yes, it can be done, but now all three silhouettes must be satisfied at once. Think of it as a switch for each little cube position (x across, y deep, z high): put a cube there only if x–z is filled in the C, x–y is filled in the H and y–z is filled in the F. Then fill in whatever extra cubes are needed to make each of the three silhouettes complete.

(iv) Other combinations. Any three letters that agree on the shared dimensions and whose shadows can be met at the same time. Easy ones to try:

• L, L, L — three L-shaped views; a natural corner shape does this.
• T, T, I — a T-shaped slab standing in a T-shaped plan.
• I, O, I — a plain cuboid gives simple rectangular views.
• Letters made of straight strokes (E, F, H, I, L, T) are much easier than round ones (O, S, C-with-curves), because a silhouette made of cubes can only have square corners.

Reconstruct the solid from the three views instead of guessing, then compare with the seven pictures.

Step 1 — read the sizes. The top view is 2 squares wide and 3 squares deep, the front view is 2 wide and 3 tall, and the side view is 3 deep and 3 tall. So the solid fits in a box 2 wide × 3 deep × 3 high, and the shared dimensions agree.

Step 2 — write the plan. The top view is an L: both cells of the back row and both of the middle row are filled, but in the front row only the left-hand cell is filled.

Step 3 — put in the heights. The side view says that exactly one of the three depths reaches 3 cubes high and the other two are only 1 cube high; and since that tall part is at one end of the side view, it is the back row. The front view says the left column reaches 3 and the right column reaches 2. Both of these can only happen in the back row, so:

So the solid is: a flat L-shaped base one cube high (two cells across the back, two across the middle, one at the front left), with two more cubes stacked on the back-left cell and one more on the back-right cell. It is the picture that shows a three-cube-high step at the back left, a two-cube-high column beside it, and a single layer running forward — and no slot or hole cut into the front, since our plan has none.

Only one view is given in each part, and one view never fixes a solid. So there are many correct answers — the task is to build any solid with the given silhouette.

The easy general recipe. Whatever the given shape is, build it one cube thick:

So for parts (i), (iv) and (vii), which give top views, lay the cubes flat on the table in exactly the pattern shown. For (ii), (v) and (viii), which give front views, stand a one-cube-thick wall up in that pattern. For (iii), (vi) and (ix), which give side views, do the same but facing sideways.

20 cubes.

Count what you can see first: the front row has 4 cubes, the row above it 3, then 2, then 1 at the top — that is 10 visible cubes. But each of those upper rows is set one step further back as well as one step higher, so nothing is holding them up unless there are cubes hidden underneath. Those hidden cubes must be counted too.

Work depth by depth, from the front of the stack backwards:

Three families of shapes: a square, a rectangle, and a hexagon — including a regular hexagon.

Why the hexagon comes out regular. Stand the cube on one corner so that the body diagonal is vertical. Turning the cube through 120° about that diagonal sends each of the three edges at the bottom corner to the next one — so the three of them are interchangeable, and their projections must be equal in length and spaced 120° apart. With an edge of 1 unit: