NCERT Solutions for Class 8th Maths Chapter 4 Exploring Some Geometric Themes
Updated on 2026-09-19
About this chapter
A fractal is a self-similar shape: it contains smaller copies of itself, over and over. Ferns, trees, coastlines, lightning and the towers of the Kandariya Mahadev Temple at Khajuraho all show this. The Sierpinski Carpet keeps 8 of every 9 sub-squares at each step, so R n = 8 n squares remain, H n = (8 n − 1)⁄7 holes have appeared, and the remaining area is (8⁄9) n — shrinking towards 0. The Sierpinski Gasket keeps 3 of every 4 triangles: 3 n triangles remain, (3 n − 1)⁄2 holes have been cut, and the area left is (3⁄4) n . The Koch Snowflake replaces every side by four sides one-third as long. So it has 3 × 4 n sides of length (1⁄3) n , and a perimeter 3 × (4⁄3) n that grows without limit — an unbounded boundary around a bounded region. A net is a solid unfolded flat. A cube has 11 nets, a
- .1 Fractals
- Sierpinski Gasket
- Koch Snowflake
- .2 Visualising Solids
- Making Solids
- Nets of a cube
- Nets of other solids
- Nets of a cylinder and a cone
- The octahedron and the dodecahedron
- Shortest Paths on a Cube
Quick revision
| Idea | Rule for one step | Formula at step n | What it tells you |
|---|---|---|---|
| Sierpinski Carpet — squares left | Rn+1 = 8 Rn | Rn = 8n | 1, 8, 64, 512, … |
| Sierpinski Carpet — holes | Hn+1 = Hn + Rn | Hn = (8n − 1)⁄7 | 0, 1, 9, 73, … |
| Sierpinski Carpet — area | each square shrinks to 1⁄9 | (8⁄9)n | Area falls to 0; the fractal has no area |
| Sierpinski Gasket — triangles left | Tn+1 = 3 Tn | Tn = 3n | 1, 3, 9, 27, … |
| Sierpinski Gasket — holes | Hn+1 = Hn + Tn | Hn = (3n − 1)⁄2 | 0, 1, 4, 13, … |
| Sierpinski Gasket — area | each triangle shrinks to 1⁄4 | (3⁄4)n | Area falls to 0 |
| Koch Snowflake — sides | each side becomes 4 sides | Sn = 3 × 4n | 3, 12, 48, 192, … |
| Koch Snowflake — perimeter | length × 4⁄3 each step | Pn = 3 × (4⁄3)n | 3, 4, 16⁄3, … grows without bound |
| Prism on an n-gon | — | n + 2 faces, 3n edges, 2n vertices | F + V − E = 2 |
| Pyramid on an n-gon | — | n + 1 faces, 2n edges, n + 1 vertices | F + V − E = 2 |
| Nets | unfold the solid flat | cube 11, tetrahedron 2, octahedron 11, dodecahedron 43,380 | A sphere has no net |
| Shortest path on a cuboid | unfold, then join by a straight line | check every unfolding | Different unfoldings give different lengths |
| Projection of a segment | drop a perpendicular to the plane | p ≤ l, with p = l only when the segment is parallel to the plane | Projections shorten, never stretch |
| Three standard views | project on three perpendicular planes | front view, top view, side view | One view alone does not fix the solid |
Exercises
- .1 Fractals — In-text Questions Page 714
- .1 Fractals — In-text Question Page 724
- Sierpinski Gasket — Figure it Out Page 72
- Koch Snowflake — Figure it Out Page 73
- .2 Visualising Solids — Build it in Your Imagination Page 75–774
- Making Solids — In-text Questions Page 79
- Making Solids — In-text Questions Page 80
- Nets of a cube — Figure it Out Page 80–81
- Nets of other solids — In-text Questions Page 81
- Nets of a cylinder and a cone — In-text Questions Page 82
- The octahedron and the dodecahedron — In-text Questions Page 83
- Shortest Paths on a Cube — In-text Questions Page 84–85
- Shortest Paths on a Cube — In-text Questions Page 86–87
- Representation of Solids on a Plane Surface — In-text Questions Page 89–91
- Front view, top view and side view — Figure it Out Page 92–93
- Shadows — In-text Questions Page 94
- Views of solids built from cubes — Figure it Out Page 95–97
- Isometric Projections and isometric drawing — In-text Questions Page 97–100
- Drawing on Isometric Grids — Figure it Out Page 100–102