Each of our Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 4 Exploring Some Geometric Themes Worksheet with Answers focuses on conceptual clarity.
Exploring Some Geometric Themes Worksheet Class 8
Class 8 Maths Exploring Some Geometric Themes Worksheet
Exploring Some Geometric Themes Class 8 Ganita Prakash Worksheet
Fractals
Rohan notices that a small fern leaf looks just like the whole plant and starts wondering why the same shape keeps repeating.
Curious, he shares his thought with his sister Riya, and they talk about how nature often creates beautiful patterns by repeating the same design again and again.
Question 1.
Write the names of any five examples of fractal patterns that you can observe in nature, art, or buildings around you.
Answer:
Question 2.
In the Sierpinski Triangle, how many new smaller triangles are formed in step 3, considering step 0 begins with one triangle? Explain the pattern.
Answer:
27
Question 3.
Find a general expression for the number of small triangles remaining after the nth step in the Sierpihski Triangle sequence.
Answer:
3n
Question 4.
In the Sierpihski Carpet, starting with one square at step 0, how many squares are removed in step 4?
Answer:
512
Question 5.
Which fractal decreases faster in area Sierpihski Triangle or Sierpihski Carpet? Justify your answer.
Answer:
Sierpinski Triangle
Shortest paths on a cube
Question 6.
A bug wants to move from one corner of a cube to the diagonally opposite corner on the other face. Explain the method to find the shortest path on the surface of the cube.
Answer:
Visualising Solids
Question 7.
A solid shows a rectangular shape when seen from the front and a circular shape when seen from the top. Name the solid.
Answer:
Cylinder
Question 8.
If the front view of a solid is a square, can there be more than one solid that gives this same view? Explain.
Answer:
Question 9.
Name the different shapes that can be seen when a cone is viewed from different directions.
Answer:
Top view: Circle;
front view: Triangle
Side view: Triangle
Question 10.
Identify the correct net(s) of a cube from the following nets.
Answer:
(a), (b)
Question 11.
A cylinder has a diameter of 8 cm and a Length of 12 cm. Which of the foLLowing is its correct net?
Answer:
Question 12.
Circle the correct net for each of the following solids. One has been. done for sou.
Answer:
Representation of Solids on a Plane Surface
Question 13.
Draw the front, top, and side views of the following solids.
Answer:
Question 14.
Circle the correct solid for the given respective views. One has been done for you.
Answer:
Question 15.
Draw solids for the given respective views.
Answer:
Isometric Projections
Question 16.
In an isometric drawing of a solid, what happens to the parallel edges of the solid, and why?
Answer:
Drawing on Isometric Grids:
Isometric grids are used to draw three-dimensional solids on paper. The grid shows three directions, representing length, depth, and height. While drawing on an isometric grid, it is helpful to draw one edge at a time, counting the number of units along each direction. Begin with simple solids like a 1 × 1 × 1 cube, then move on to larger solids such as a 2 × 2 × 2 cube. Lightly sketch the figure first and darken only the visible edges to clearly show the solid.
Answer:
Question 17.
Draw the following solids on the isometric grid paper.
Answer:
Worksheet On Exploring Some Geometric Themes Class 8
A. Choose the correct option.
1. Which of the following is true about the Sierpinski Triangle?
(a) It is formed by joining the midpoints of an equilateral triangle to create 4 smaller equilateral triangles.
(b) It is formed by cutting an equilateral triangle into two smaller triangles.
(c) It is formed by joining the corners of an equilateral triangle.
(d) It is formed by dividing an equilateral triangle into two right-angled triangles.
Answer:
(a) It is formed by joining the midpoints of an equilateral triangle to create 4 smaller equilateral triangles.
2. What Is true for the Koch Snowflake?
(a) It is formed by adding smaller triangles to a square.
(b) It is a fractal that begins with an equilateral triangle.
(c) It is created by doubling the size of a square.
(d) It has no self-similarity.
Answer:
(b) It is a fractal that begins with an equilateral triangle.
3. Which of the following Is an example of a fractal pattern found In nature?
(a) A straight road
(b) A leafs vein pattern
(c) A smooth flat surface
(d) A plain wall
Answer:
(b) A leafs vein pattern
4. A prism has congruent polygons with 10 sides. How many edges does the prism have?
(a) 20
(b) 30
(c) 40
(d) 10
Answer:
(b) 30
5. Which of the following statements is correct regarding nets of a cube?
(a) There are 12 possible distinct nets for a cube.
(b) A cube has 11 possible net structures, considering rotations and flips.
(c) A cube has only 6 possible net structures.
(d) The nets for a cube are not affected by rotations or flips.
Answer:
(b) A cube has 11 possible net structures, considering rotations and flips.
6. To find the shortest path between two points on the surface of a cuboid, the most effective method is to:
(a) Calculate the diagonal through the centre of the cuboid using the 3D distance formula.
(b) Follow the edges of the cuboid until you reach the target face.
(c) Flatten the relevant faces into a 2D net and draw a straight line between the points.
(d) Measure the perimeter of the largest face.
Answer:
(c) Flatten the relevant faces into a 2D net and draw a straight line between the points.
Directions. (For Q.7 – 8): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false. {d) Assertion (A) is false but Reason (R) is true.
7. Assertion (A): The projection of a point onto a plane is obtained by drawing a line from the point that intersects the plane perpendicularly.
Reason (R): The projection of all points of an object onto a plane forms the overall projection of the object on that plane.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
8. Assertion (A): The size of the shadow of an object will change depending on the distance between the torch and the object.
Reason (R): As the torch moves farther from the object, the shadow becomes larger, and as the torch moves closer to the object, the shadow becomes smaller.
Answer:
(c) Assertion (A) is true but Reason (R) is false. (d) Assertion (A) is false but Reason (R) is true.
B. Fill in the blanks:
1. By joining the midpoints of an equilateral triangle, we divide it into ______ identical equilateral triangles.
Answer:
four
2. The Koch Snowflake is a type of ______ fractal, which means its structure repeats at every smaller scale.
Answer:
self similar
3. The number of faces of a prism with congruent polygons having n sides is ______.
Answer:
n + 2
4. A cube has ______ distinct net structures when rotations and flips are considered equivalent.
Answer:
11
C. State whether the following statements are true (T) or false (F).
1. The Sierpinski Triangle is created by joining the midpoints of an equilateral triangle, which divides it into 4 smaller equilateral triangles.
Answer:
True
2. The Koch Snowflake has a finite perimeter but an infinite area.
Answer:
False
3. A prism with a polygon having n sides will always have 2n vertices.
Answer:
True
4. Two cube nets are considered different if one can be obtained from the other by rotating or flipping.
Answer:
False
5. The shortest path for an ant to reach a laddu placed on a cuboidal box will always involve travelling along the physical edges of the cuboid.
Answer:
False
D. Solve the following.
Question 1.
Explain how joining the midpoints of an equilateral triangle creates 4 identical equilateral triangles and describe the resulting figure.
Answer:
Question 2.
Explain the process of constructing the Koch Snowflake starting with an equilateral triangle.
Answer:
Question 3.
If a prism has congruent polygons with n sides, how would you calculate the number of faces, edges, and vertices of the prism?
Answer:
Faces = n + 2
Edges = 3n
Vertices = 2n
Question 4.
A cuboidal box has a length of 6 cm, a width of 3 cm, and a height of 3 cm. An ant is at the bottom-left corner of the front face, and a laddu is placed at the top-right corner of the opposite back face. Find the shortest path the ant must travel on the surface of the box to reach the laddu.
Answer:
6√2 cm