Class 8 Mathematics

Chapter 11 Solutions — Exploring Some Geometric Themes

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Solutions

Overview

Step-by-step NCERT solutions for Exploring Some Geometric Themes (Chapter 11, NCERT Class 8 Mathematics) — the full working for every question, not just the final answer. You can also read the Exploring Some Geometric Themes textbook chapter.

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What these solutions cover

All 12 questions in Exploring Some Geometric Themes are solved in the PDF. Here's what's inside, exercise by exercise:

Sierpinski Gasket

Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.
Find the area of the region remaining at the nth step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.

Koch Snowflake

Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.
Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.

Nets of a Cube

1.
Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try. (Six hexomino shapes labelled
- (i) to
- (vi) are shown.)
2.
Draw a net of a cuboid having sidelengths:
- (i) 5 cm, 3 cm, and 1 cm
- (ii) 6 cm, 3 cm, and 2 cm

Projections

Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?
Match each of the following objects with its projections (front, top, side views). Objects shown: mug, funnel, hammer, car, slide/ramp, chair, ceiling fan, tin can.

Views of Cube Combinations and Shadows

Which solid corresponds to the given top view, front view, and side view? (Front view: C-shape / staircase; Top view: offset staircase from above; Side view: step/small block.)
Find the number of cubes in this stack of identical cubes. (An isometric image of a stepped pyramid arrangement is shown.)

Drawing on Isometric Grids

In addition to the 5 ways shown in Fig. 4.8, are there any additional ways of gluing four cubes together along faces? Can you visualise and draw these as well?
2.
Observe this impossible triangle (Penrose triangle made of cubes).
- (i) Would it be possible to build a model out of actual cubes? What are the front, top, and side profiles of this impossible triangle?
- (iii) Why does the illusion work?

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