Opening questions…

Exploring Some Geometric Themes

Enable JavaScript to open the explanations.

Page 71: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.
Page 71: By its construction, each step in the sequence has (i) squares of the same size that remain in the figure, and the…
Page 71: Similarly, how do we find the number of holes at a given step? Every square that remains at the nth step gives rise…
Page 72: Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.
Page 72: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.
Page 72: Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the…
Page 72: Find the area of the region remaining at the nth step in each of the shape sequences that lead to the Sierpinski…
Page 73: Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake.
Page 73: Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.
Page 73: Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a…
Page 75: Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see…
Page 76: 2. Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What…
Page 77: Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can…
Page 77: Can you visualise solids that have the following contrasting profiles? 8. A solid with a rectangular profile from…
Page 79: If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if…
Page 79: If the base of a pyramid has 10 sides, how many faces, edges and vertices does the pyramid have? What if the base is…
Page 80: Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try.
Page 81: A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be…
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
Page 81: What is a net of a regular tetrahedron? Which of the following are nets of a regular tetrahedron? Are there any…
Page 81: Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making…
Page 82: What is the net of a cylinder? If the circular faces of a cylinder are unfolded, and if a cut is made along the…
Page 82: How will the net of a cone look? If the cone is slit open along the line l and then unrolled, what will we get?
Page 82: What surface do you construct by using the above net, in which O is not the centre of the boundary circle? Make a…
Page 83: Can you visualise its net? Taking all the triangles in the net to be equilateral, make a cutout of the net and fold…
Page 83: Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without…
Page 84: What is the shortest path for the ant to reach the laddu? What about if the laddu is instead at the centre of an…
Page 85: What does this show?
Page 86: Find the shortest path between the ant and the laddu in the following case, where the laddu is 2 cm from one bottom…
Page 87: What is the length of the shortest path between the ant and the laddu?
Page 88: Let us visualise the projection of a line.
Page 89: Can you now compare the lengths p and l? When is the length of the projected line equal to its actual length?
Page 89: What do you think are the different possible projections of a square that we get based on its orientation? What do…
Page 90: See the earlier figures of projected lines, cubes, and cones. In each case, see if you can visualise another object…
Page 91: The projection on the vertical plane is called the front view, on the horizontal plane the top view, and on the side…
Page 92: Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between…
Page 92: Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to…
Page 93: Match each of the following objects with its projections (front, top and side views): a mug, a funnel, a hammer, a…
Page 94: What do you see? Observe what happens to the size of the shadow as you vary the distance between your torch and your…
Page 95: Draw the top view, front view and side view of each of the following combinations of identical cubes.
Page 95: (i) This looks like a 'C' from the front. What does it look like from the side? From the top? (ii) Glue additional…
Page 96: Which solid corresponds to the given top view, front view, and side view?
Page 96: Using identical cubes, make a solid that gives the following projections.
Page 96: Find the number of cubes in this stack of identical cubes.
Page 96: What are the different shapes the projection of a cube can make under different orientations?
Page 97: Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand…
Page 98: Why is this correspondence between directions on isometric paper and axes of the solid so effective for…
Page 100: In addition to the 5 ways shown (all flat, one-layer arrangements), are there any additional ways of gluing four…
Page 101: Draw the following figures on the isometric grid.
Page 101: Is there anything strange about the path of this ball? Recreate it on the isometric grid.
Page 101: Observe this triangle. (i) Would it be possible to build a model out of actual cubes? What are the front, top, and…

Back to Mathematics