Fractals and Visualising Solids | FIO

Question 17

Find the number of cubes in this stack of identical cubes.

Solution

We can find the total number of cubes by counting the cubes in each horizontal layer and then adding them together.

Step 1 — Count cubes in each layer

Let us count the cubes starting from the top layer and moving downwards.

The topmost layer has only one cube.

$\text{Cubes in Layer 1} = 1$

The second layer from the top supports the cube above it. We can see it forms a small triangle.

$\text{Cubes in Layer 2} = 3$

The bottom layer supports the cubes in the second layer. We can count them by rows. The front row has 3 cubes. The row behind it has 2 cubes. The last row has 1 cube.

$\text{Cubes in Layer 3} = 3 + 2 + 1$

$= 6$

$\boxed{\text{Cubes in Layer 3} = 6}$

Diagram 1

Step 2 — Find the total number of cubes

Now, we add the number of cubes from all three layers to find the total.

$\text{Total cubes} = \text{Cubes in Layer 1} + \text{Cubes in Layer 2} + \text{Cubes in Layer 3}$

$= 1 + 3 + 6$

$\boxed{10}$

Answer

(i) The total number of cubes in the stack is 10.

More questions in FIO

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.

Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.

Find the area of the region remaining at the $n$ th step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake.

Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.

Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.

Figure it Out

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

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.

Draw a net of a cuboid having sidelengths:

(i) 5 cm, 3 cm, and 1 cm

(ii) 6 cm, 3 cm, and 2 cm

Q10

Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?

Q11

Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to the vertical, horizontal and side planes: cube, cuboid, parallelepiped, cylinder, cone, prism, and pyramid. If needed, see the next problem for clues.

Q12

Match each of the following objects with its projections.

Q13

Draw the top view, front view and the side view of each of the following combinations of identical cubes.

Q14

Imagine eight identical cubes, glued together along faces to form the letter 'C'.

Q15

Which solid corresponds to the given top view, front view, and side view?

Solids to choose from:

Q16

Using identical cubes, make a solid that gives the following projections:

Q17

Find the number of cubes in this stack of identical cubes.

Q18

What are the different shapes the projection of a cube can make under different orientations?

Q19

In addition to the 5 ways shown in Fig. 4.8, are there any additional ways of gluing four cubes together along faces? Can you visualise and draw these as well?

Q20

Draw the following figures on the isometric grid.

[Hint: It may be useful to determine whether the edge to be currently drawn — say, along the height — goes from down to up or up to down. Accordingly, draw the line segment on the grid either in the direction of the height axis or opposite to it.]

Q21

Is there anything strange about the path of this ball? Recreate it on the isometric grid.

[Hint: Consider a portion of this figure that is physically realisable and identify the 3 primary directions.]

Q22

Observe this triangle.

(i) Would it be possible to build a model out of actual cubes? What are the front, top, and side profiles of this impossible triangle? (ii) Recreate this on an isometric grid. (iii) Why does the illusion work?