Question 13
Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.
A net is a flat shape that can be folded to make a three-dimensional object.
Step 1 — Understanding a Square Pyramid Net
Let us understand what a square pyramid is. It has a square base at the bottom. It also has four triangular faces. These four triangles meet at a single point called the apex. A net for a square pyramid will show all these faces laid out flat. It will consist of one square and four triangles. The four triangles will be attached to the sides of the square.
Step 2 — Choosing Appropriate Measurements
We need to choose a size for our square pyramid. Let us choose the side length of the square base. We will use a side length of 4 cm for the square. The four triangular faces must be isosceles triangles. This means two sides of each triangle are equal in length. The base of each triangle will be 4 cm, matching the square's side. We need to decide on the length of the two equal sides of these triangles. Let us choose these equal sides to be 6 cm long. This length will be the slant edge of our pyramid.
Step 3 — Drawing the Net
First, draw a square in the center of your paper. Its side length should be 4 cm. Now, on each of the four sides of this central square, draw an isosceles triangle. For each triangle, its base is the 4 cm side of the square. To draw one such triangle, place your compass point at one end of the 4 cm base. Open the compass to a radius of 6 cm. Draw an arc above the base. Repeat this from the other end of the 4 cm base. The point where the two arcs intersect is the top corner (apex) of the triangle. Connect this point to both ends of the 4 cm base. Repeat these steps for all four sides of the central square. You will now have a central square with four triangles attached to its sides.
Step 4 — Verifying the Net
To verify if the net works, we will perform a simple cutout and fold. Carefully cut out the entire shape you have drawn. Cut along all the outer edges of the net. These are the outer sides of the four triangles. Now, gently fold the paper along the inner lines. These inner lines are the sides of the central square. As you fold, the four triangular faces will rise upwards. The top corners of these four triangles will meet perfectly at a single point. This point is the apex of the pyramid. This action forms a complete square pyramid.
Answer
(i) To draw the net, first draw a central square with a side length of 4 cm. On each of the four sides of this square, draw an isosceles triangle. Each isosceles triangle should have a base of 4 cm (matching the square's side) and two equal sides of 6 cm. (ii) To verify, cut out the entire net along its outer edges. Fold the four triangles upwards along the sides of the central square. The top corners of the four triangles will meet at a single point, forming the apex of the square pyramid.
More questions in IT
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.
By its construction, each step in the sequence has (i) squares of the same size that remain in the figure, and the size of these squares becomes smaller and smaller as the step number increases, and (ii) square holes that are formed by removing square pieces.
Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.
[Hint: Note that the corner triangles are isosceles.]
This fractal is called the Sierpinski Triangle/Gasket.
In previous classes, you've seen solids that are much simpler than an elephant or cat, such as cubes, spheres, cylinders, and cones. What would the profiles of these look like, from different viewpoints?
Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can imagine the solid passing through a wall like Tom did, and leaving a hole of the appropriate shape.
- A solid whose profile has a square outline
- A solid whose profile has a circular outline
- A solid whose profile has a triangular outline
As we saw with the elephant, a given solid might have very different profiles from different viewpoints. Can you visualise solids that have the following contrasting profiles?
Spend some time on this, and if you are finding it difficult to visualise, you may look around and use objects that are around you, or that you will make in the next section. Feel free to consider viewpoints from any direction, including directly above the object.
- A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint
- A solid with a circular profile from one viewpoint and a triangular one from another viewpoint
- A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint
- A solid with a trapezium shaped profile from one viewpoint and a circular one from another viewpoint
- A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint
Are there unique solids for each of the conditions, or can you come up with multiple possibilities?
If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if the polygons have sides?
If the base of a pyramid has 10 sides, how many faces, edges and vertices does the pyramid have? What if the base is an -sided polygon?
What is a net of a cube?
Visualise how it can be folded to form a cube.
What is a net of a regular tetrahedron? Which of the following are nets of a regular tetrahedron?
Are there any other possible nets?
Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making an actual cutout.
Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.
What is the net of a cylinder?
If the circular faces of a cylinder are unfolded, and if a cut is made along the height of the cylinder, as shown in the figure below, then we get
What are the sidelengths of the rectangle obtained?
How will the net of a cone look?
If the cone is slit open along the line and then unrolled, what will we get?
Observe that all the points on the boundary of the base circle are at equal distances from 'O'. So after unrolling the cone, the boundary of the net will be a portion of a circle with centre O.
What surface do you construct by using the above net, in which O is not the centre of the boundary circle? Make a physical model to help you answer this question!
Draw a net with appropriate measurements that can be folded into a triangular prism. Verify that it works by making an actual cutout.
Taking all the triangles in the net to be equilateral, make a cutout of the net and fold it to form an octahedron.
What is the shortest path for the ant to reach the laddu?
What about in the following case?
If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities?
For example, are either of these the shortest path?
What does this show?
Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?
What is the length of the shortest path between the ant and the laddu?
What happens to the length of a line in its projection?
Can you now compare the lengths and ?
When is the length of the projected line equal to its actual length?
What do you think are the different possible projections of a square that we get based on its orientation?
What do you think is the projection of a parallelogram under different orientations?
Can this ever be a quadrilateral that is not a parallelogram? As a starting point, you could think about the projection of a pair of parallel lines.
What can you say about the projection of an -sided regular polygon?
[Hint: Projection of a polygon is composed of the projections of its sides.]
How would the projections of a cube and a cone look?
See Figures 4.2–4.5. In each case, see if you can visualise another object that gives the same projection.
Find another object that makes the same projection as that of a given cone.
Have you played Tetris? There are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.
Context: In Fig. 4.8, there are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.
Q. Imagine these are cubes, not squares. Draw each of these on your isometric paper (you can find it at the end of the book).
Why is this correspondence between directions on isometric paper and axes of the solid so effective for communicating the shape of the solid?
Can you try drawing the other tetris shapes on isometric paper?