New NCERT Ganita Prakash Part 2 · Class 8 Maths · Part II · Chapter 4: Exploring Some Geometric Themes
This chapter explores fractals, solid shapes and their nets, shortest paths on a cube and views of solids. It gives counting rules for fractals and for the faces, edges and vertices of prisms and pyramids.
From the Class 8 Maths Formula Sheet, made as per the new NCERT book Ganita Prakash · Pages 45–48 of the book.
Key points
- A fractal is a self-similar shape where the same pattern repeats at smaller and smaller scales.
- Fractals in nature include ferns, trees, clouds, coastlines and lightning.
- The Sierpinski carpet, Sierpinski triangle and Koch snowflake are made by repeating a simple rule.
- The Kandariya Mahadev Temple, Khajuraho (c. 1025 CE) is perhaps among the oldest human-made fractals.
- A cube or cuboid has 6 faces, 12 edges and 8 vertices.
- A net is a flat shape that folds into a solid; a cube has 11 different nets.
- A cylinder net is 2 circles and a rectangle; a cone net is a sector of a circle and a circle.
- The shortest path on the surface of a cube or cuboid is a straight line on a suitable net.
- Front, top and side views are used because one view alone does not fix the solid.
- The isometric view of a cube is a regular hexagon.
Formulas
| Prism with n-sided base | F = n + 2, E = 3n, V = 2n |
| Pyramid with n-sided base | F = n + 1, E = 2n, V = n + 1 |
| Sierpinski carpet: squares left at step n | Rₙ = 8ⁿ Rₙ₊₁ = 8Rₙ, R₀ = 1 |
| Sierpinski carpet: holes | Hₙ₊₁ = Hₙ + Rₙ H₀ = 0 → 0, 1, 9, 73 … |
| Sierpinski carpet: area left | (8/9)ⁿ start area 1 |
| Sierpinski triangle | triangles left = 3ⁿ ; area left = (3/4)ⁿ |
| Koch snowflake: number of sides | 3 × 4ⁿ perimeter × 4/3 each step |
Key terms
- Fractal: A self-similar shape in which the same pattern repeats at smaller and smaller scales.
- Net: A flat shape that folds into a solid.
- Faces, edges, vertices: Flat surfaces, line segments and corners of a solid.
- Tetrahedron: A triangular pyramid.
- Isometric: "Equal measure": all edges of a cube are drawn with equal lengths.
- Projection: The view of a solid on a plane; a line's projection is at most its actual length.
Diagrams to learn
- Steps 0 to 3 of the Sierpinski carpet, Sierpinski triangle and Koch snowflake
- Prism and pyramid with faces, edges and vertices
- Front, top and side views, and a cube on an isometric grid
- Table of faces, edges and vertices of solids
- Fractal counts table
- Flaps are not part of the net.
- Prism faces other than the two ends are parallelograms; pyramid side faces are triangles meeting at the apex.
- Check every way of unfolding for the shortest path — a line that leaves the net is not allowed.
- One view alone does not fix the solid, so we use three views.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 8 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is a fractal in Class 8 Maths?
A fractal is a self-similar shape where the same pattern repeats at smaller and smaller scales. Examples are the Sierpinski carpet, the Sierpinski triangle and the Koch snowflake.
How many faces, edges and vertices does a prism have?
A prism with an n-sided base has n + 2 faces, 3n edges and 2n vertices. A triangular prism has 5 faces, 9 edges and 6 vertices.
How many nets does a cube have?
A cube has 11 different nets.
← Part II · Chapter 3: Proportional Reasoning – 2Part II · Chapter 5: Tales by Dots and Lines →
