Question 7
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 net of a cube is a 2D pattern of connected squares that can be folded along their edges to form a closed 3D cube without any overlaps or missing faces.
- To check if a figure is a valid net:
- Choose one square as the base (bottom face).
- Fold the adjacent squares upwards to form the sides (front, back, left, right).
- Fold the remaining square over the top.
- If every face is covered exactly once with no overlaps, the net is valid.
(i) Is Figure (i) a net of a cube?
Step 1 · Analyze Folding of Figure (i)
Labeling the squares:
- Let be the base.
- , , and fold up as the left, right, and back faces.
- Folding and results in both squares overlapping or failing to meet at the required top-front-left edges, leaving one side open.
Therefore, this figure cannot form a cube.
(i) No
(ii) Is Figure (ii) a net of a cube?
Step 1 · Analyze Folding of Figure (ii)
Labeling the squares:
- is attached to , and is attached to .
- When folded with any chosen base (e.g., or ), two faces overlap on the same side, leaving the opposite side open.
Therefore, this figure cannot form a cube.
(ii) No
(iii) Is Figure (iii) a net of a cube?
Step 1 · Analyze Folding of Figure (iii)
Labeling the squares:
- Base:
- Left face:
- Right face:
- Back face:
- Front face: (folded from )
- Top face: (folded from )
All faces cover the cube without gaps or overlaps.
(iii) Yes
(iv) Is Figure (iv) a net of a cube?
Step 1 · Analyze Folding of Figure (iv)
Labeling the squares:
- Base:
- Left face:
- Right face:
- Back face:
- Front face:
- Top face: (folded from )
All faces cover the cube without gaps or overlaps.
(iv) Yes
(v) Is Figure (v) a net of a cube?
Step 1 · Analyze Folding of Figure (v)
Labeling the squares:
- Base:
- Left face:
- Right face:
- Back face:
- Front face: (folded from )
- Top face: (folded from )
All faces fold into a complete cube.
(v) Yes
(vi) Is Figure (vi) a net of a cube?
Step 1 · Analyze Folding of Figure (vi)
Labeling the squares:
- Base:
- Left face:
- Right face:
- Back face: (folded from )
- Front face:
- Top face: (folded from )
All faces fold into a complete cube.
(vi) Yes
- Assuming 6 squares always form a cube: Having exactly 6 squares is necessary but not sufficient; their arrangement determines if faces will overlap.
- Overlapping adjacent flaps: If two flap squares are attached on the same side of a row directly adjacent to each other (like in figure (ii)), they will fold into the same position and overlap.
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 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 step of the shape sequence that leads to the Koch Snowflake.
Find the perimeter of the shape at the 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) , , and
(ii) , , and
Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?
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.
Match each of the following objects with its projections.
Draw the top view, front view and the side view of each of the following combinations of identical cubes.
Imagine eight identical cubes, glued together along faces to form the letter 'C'.
Which solid corresponds to the given top view, front view, and side view?
Solids to choose from:
Using identical cubes, make a solid that gives the following projections:
Find the number of cubes in this stack of identical cubes.
What are the different shapes the projection of a cube can make under different orientations?
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?
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.]
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.]
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?