Students often refer to Class 8 Maths Notes and Part 2 Chapter 4 Exploring Some Geometric Themes Class 8 Notes during last-minute revisions.
Class 8 Maths Chapter 4 Exploring Some Geometric Themes Notes
Class 8 Exploring Some Geometric Themes Notes
→ Fractals are self-similar geometric objects found in nature and in art.
→ The Sierpinski Carpet, Sierpinski Gasket, and Koch Snowflake are some examples of mathematical fractals. They can be obtained by repeatedly applying certain geometric operations, which generate a sequence of shapes that approach the fractal.
→ Cuboids, tetrahedra, cylinders, cones, prisms, pyramids, and octahedra are some of the solids that can be obtained by folding suitable nets.
→ The shortest path between two points on the surface of a cuboid can be found by using a suitable net of the cuboid.
→ Any object can be represented on a plane surface by using its projections on plane surfaces. For this purpose, we generally use the front view (projection onto the vertical plane), top view (projection onto the horizontal plane), and side view (projection onto the side plane) of the object.
→ A cube can be oriented such that the lengths of all its edges in the projection are equal. Such a projection is called the isometric projection. Isometric projections of different solids can be drawn using isometric grid paper.
Fractals
Fractals are complex, never-ending patterns that are similar and repetitive in nature.
Fractals are formed when a pattern is recursively repeated within itself at different scales.
A tree is an example of a fractal—like structure; it shows the pattern seen in the tree itself.
The repeated branching pattern from the tree to its branches and then to twigs helps in maximizing the area for the growth of leaves at different scales.
Sierpinski Carpet
The Sierpinski carpet was discovered by polish mathematician Sierpinski.
How to make/understand a Sierpinski carpet?
Step 0: Draw a square.
Step 1: Divide it into 9 small squares and remove the central square.
Step 2: Divide each of the eight squares around the central square and divide each of them into 9 smaller parts.
Remove the central square of all eight squares.
Keep repeating the pattern, and thus we get a Sierpinski carpet.
Now, let us calculate the total number of holes in n squares.
Step 0: 0 hole
Step 1: 1 hole
Step 2: 1 + 8 = 9 holes
Step 3: 1 + 8 + 8 × 8 = 1 + 8 + 64 = 73 holes
Step 4: 1 + 8 + 64 + 8 × 8 × 8 = 1 + 8 + 64 + 512 = 585 holes
We can also write it as 81-1 + 82-1 + 83-1 + 84-1
For step n: 81-1 + 82-1 + 83-1 + 84-1 + … + 8n-1
Now we calculate the number of squares left after a square is removed.
Step 0: 1 square
Step 1: 8 squares
Step 2: 8 × 8 = 64 squares.
Step 3: 8 × 8 × 8 = 512 squares
.
.
.
For step n: 8n
Sierpinski’s Gasket
A Fractal is formed by repeatedly removing the central triangle from an equilateral triangle, leaving three smaller triangles at each step. It shows self-similarity, as each small triangle resembles the whole.
Koch Snowflake
The Koch Snowflake is a fractal made by repeatedly adding a triangular bump on each side of a triangle. It shows self-similarity.
Visualising Solids
In our daily life, we see many objects around us like books, balls, ice-cream bricks, ice-cream cones, a ludo dice, etc., which have different shapes.
They all have one thing (feature) in common— they all occupy space. We can see, touch, and feel them from all sides.
In other words, they all have three dimensions—length, breadth, and height. Hence, they are called three-dimensional (3-D) shapes (that is, solid shapes).
The objects mentioned above are all 3-D (solid shapes).
Shapes such as triangles, rectangles, squares, circles, etc. are 2-dimensional (2-D) figures. They have length and breadth.
We can draw them on a plane (paper sheet, blackboard), but we can’t hold them as they have no height. These shapes are called plane figures as they can be drawn on a plane.
Figures having length only are known as one-dimensional figures.
For example, a line is a one-dimensional figure.
Visualising solids means understanding and picturing three-dimensional (3-D) shapes in our minds.
Making Solids
A solid is a three-dimensional figure that has length, breadth, and height. Solids occupy space and have a definite shape.
While making or studying solids, we focus on three components, namely faces, edges, and vertices.
(a) Face: Face is a flat or curved surface. A flat face will have a boundary, but a curved face has no boundary.
Examples:
1. A solid cylinder has two flat surfaces, i.e., the circular faces and one curved face.
2. A cube has 6 flat faces.
(b) Edge: Edge is a straight line segment or a curved line where two faces meet.
Example:
A cube has 12 edges and a cylinder has 2 edges (both curved).
(c) Vertex: A vertex is a point in the solid figure. The plural form of vertex is vertices. A vertex is also called the corner.
Example:
A cone has one vertex.
A cube has eight vertices.
A cylinder has no vertex.
Special Solids
Prism: A prism is a solid with two congruent polygonal faces placed parallel to each other, and the corresponding edges are connected with rectangles or parallelograms.
The name of the prism depends on the name of the congruent polygons.
Pyramids: A pyramid is a solid with a polygonal face, and all vertices of the polygon are connected to a point (not on the face) to make triangular faces.
The name of the pyramid depends on the name of the polygonal base.
Mathematician Leonhard Euler gave Euler’s formula, which established the relation between faces, edges, and vertices of a polyhexagon.
Eulers Formula: F + V = 2 + E
[Learning Technique Face Value is too Easy, F + V = 2 + E]
Prism: We shall now establish a relation between the number of sides of the polygonal base of the prism and its vertices, faces, and edges.
If one of the congruent bases of the prism has n sides, then it has
Vertices = 2n
Edges = 3n
Faces = n + 2
Pyramid: We shall now establish a relation between the number of sides of the polygonal base of the pyramid and its vertices, faces, and edges.
If the polygonal base has n sides, the pyramid will have
Vertices = n + 1
Edges = 2n
Faces = n + 1
Nets of Solids
A net of a solid (3-D shape) is the outline of its faces (in 2-D, say on a piece of paper) joined together, from which a model of the solid can be made by folding.
Activity
Step 1. Take a cardboard box (3-D solid) as shown in Fig (i) below.
Step 2. Cut the edges of the box as shown in Fig (ii) below.
Step 3. Lay the box flat as shown in Fig (iii) below to get a NET (outline of the faces of the cardboard joined together) forthe given cardboard.
Moving in the reverse way, when we fold the six rectangles of the net of fig. (iii) in a suitable way, we shall get a model of the cardboard of fig. (i).
The net of a 3-D shape helps us in visualising quite a lot of details about the 3-D shape.
Regular Tetrahedron
A tetrahedron is a solid (3D) figure with four triangular faces, four vertices, and six edges. All its faces are triangles. It is also called a triangular pyramid. A net of a regular tetrahedron is four equal triangles that fold into the shape.
Regular Octahedron
An octahedron, an eight-faced polyhedron, is formed by joining two square-based pyramids at their bases.
A regular octahedron is made up of eight equilateral triangles.
Net of a regular octahedron is shown below:
By folding along the common edges, we can form an octahedron.
Dodecahedron
‘Dodeca’ means 12, so a dodecahedron has 12 faces. Each face is a regular pentagon, with 20 vertices and 30 edges.
Shortest paths on a cube
To find the shortest path between two points on a box, we need to draw the net of the box.
Now we draw a net of the box. The shortest path is a straight line path from point A to point B.
Example: A is the position of an ant on the cuboid, and B is the position of the Ladoo. Draw the net of the cuboid.
(a) When we unfold the box, it becomes a net. We see the position in 2 dimensions. The path is marked in red.
Shortest path = \(\sqrt{7^2+4^2}\)
= \(\sqrt{49+16}\)
= √65
(b) When the net is unfolded again in a different way.
The path is again marked in red.
By Baudhayana-Pythagoras Theorem the shortest path is \(\sqrt{5^2+2^2}=\sqrt{25+4}=\sqrt{29}\)
Thus, we conclude that the distance between two points will depend on the way we open the net of the cuboid.
Trick Involved
Open the net so that the adjacent faces of the positions are aligned as explained in the following example.
In the above cuboid, the shortest distance will be obtained if the net has left face, top, and right face of the cuboid are unfolded and are in a straight line.
In right ∆ABP, by Baudhayana-Pythagoras Theorem
AB2 = AP2 + PB2 = 52 + 122
⇒ AB = \(\sqrt{25+144}\)
= √169
= 13 cm
Projections
Projection is the process of representing a geometrical shape on a plane by drawing perpendicular lines from every point on the plane.
Projection of some of the geometrical shapes.
1. Point: Projection of a point is a point.
2. Line: The projection of a point can be a point or a line, depending on the angle it makes with the plane.
The shortest projection of a line that is perpendicular to the plane is a point.
The longest projection of a line is equal to the length of the line, and in this case, the line is parallel to the plane.
Shadows
A shadow play
We will need a source of light and a few solid shapes for this activity.
Shine a torch on the object perpendicular to the wall. See the figure for shadows. In shadow, the cone looks like a triangle.
A Third Way is by looking at it from Certain Angles to Get Different Views
We can look at an object standing in front of it, by the side of it, or from above.
Each time, we will get a different view. (See the following figure)
Here is an example of how we get different views of the above building.
Views of 3-D Shapes
3-D objects that are solids look different from different positions. So, they can be drawn or looked at from three different perspectives (that is, visible parts of a solid):
- Top view
- Side view
- Front view
An architect is concerned about the top view, side view, and front view.
On the internet, if we search for Google Earth, we will find pictures that are taken from satellites, and hence they are top view of the places. We can even find our town or village, our school, our sweet home, etc., in it. But they are all top views.
If we want a carpenter to make a wooden box for us, then we should draw on a paper the top view, the side view, and the front view with dimensions. Then give this paper to the carpenter so that he can make the box to our satisfaction.
Remark: But if an artist is looking at a solid, he can view it from any angle. (Slanting view or isometric view).
The solid object has four different views. (Top, side, front, and from any angle)
Isometric Grids
Drawing on an isometric grid is a popular technique for creating three-dimensional (3D) objects and scenes on a two-dimensional (2D) surface without using traditional perspective techniques or vanishing points. It is widely used in technical drawing, illustration, and game design for its clarity and consistent proportions.
The post Exploring Some Geometric Themes Class 8 Notes Maths Part 2 Chapter 4 appeared first on Learn CBSE.
📚 NCsolve - Your Global Education Partner 🌍
Empowering Students with AI-Driven Learning Solutions
Welcome to NCsolve — your trusted educational platform designed to support students worldwide. Whether you're preparing for Class 10, Class 11, or Class 12, NCsolve offers a wide range of learning resources powered by AI Education.
Our platform is committed to providing detailed solutions, effective study techniques, and reliable content to help you achieve academic success. With our AI-driven tools, you can now access personalized study guides, practice tests, and interactive learning experiences from anywhere in the world.
🔎 Why Choose NCsolve?
At NCsolve, we believe in smart learning. Our platform offers:
- ✅ AI-powered solutions for faster and accurate learning.
- ✅ Step-by-step NCERT Solutions for all subjects.
- ✅ Access to Sample Papers and Previous Year Questions.
- ✅ Detailed explanations to strengthen your concepts.
- ✅ Regular updates on exams, syllabus changes, and study tips.
- ✅ Support for students worldwide with multi-language content.
🌐 Explore Our Websites:
🔹 ncsolve.blogspot.com
🔹 ncsolve-global.blogspot.com
🔹 edu-ai.blogspot.com
📲 Connect With Us:
👍 Facebook: NCsolve
📧 Email: ncsolve@yopmail.com
😇 WHAT'S YOUR DOUBT DEAR ☕️
🌎 YOU'RE BEST 🏆