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Exploring Some Geometric Themes – Class 8 Mathematics notes, important questions and MCQs

NCERT Class 8 Mathematics, chapter 11. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Fractals are shapes in which the same pattern repeats at smaller and smaller scales, such as the Sierpinski carpet, the Sierpinski gasket and the Koch snowflake. We also learn to visualise solids: building them from nets, counting faces, edges and vertices, and drawing their views from different sides.

Fractals

Sierpinski carpet: divide a square into 9 equal squares and remove the middle one; repeat on each remaining square. After n steps, 8ⁿ squares remain and the area left is (8/9)ⁿ of the original.
Sierpinski gasket: join the midpoints of a triangle's sides and remove the middle triangle; repeat. After n steps, 3ⁿ triangles remain and the area left is (3/4)ⁿ of the original.
Koch snowflake: divide each side into three equal parts, build an equilateral triangle on the middle part and remove its base; each step multiplies the perimeter by 4/3.
Fractal patterns appear in nature (ferns, coastlines, cauliflower) and in art and architecture.

Visualising solids

A net is a flat shape that can be folded to make a solid; a cube has 11 different nets.
Polyhedra have faces, edges and vertices; Euler's formula: F + V − E = 2.
Projections (views) of a solid from the front, side and top help us draw and recognise it.

Key terms

Fractal: a shape made of parts that are similar to the whole, repeating at smaller scales
Net: a two-dimensional pattern that folds to form a three-dimensional solid
Polyhedron: a solid with flat polygonal faces, straight edges and vertices
Projection: the view of a solid seen from one direction, such as the front, side or top

Formulas

Euler's formula: F + V − E = 2
Sierpinski carpet: squares after n steps = 8ⁿ
Sierpinski gasket: triangles after n steps = 3ⁿ
Koch snowflake: perimeter × 4/3 at each step

Important questions with answers

What is a fractal? Give two examples.2 marks

A fractal is a shape made of smaller parts that look like the whole shape, with the same pattern repeating at smaller and smaller scales. Examples are the Sierpinski gasket and the Koch snowflake; in nature, ferns and cauliflower show fractal-like patterns.

Verify Euler's formula for a cube and a square pyramid.3 marks

A cube has F = 6, V = 8 and E = 12, so F + V − E = 6 + 8 − 12 = 2. A square pyramid has F = 5 (one square and four triangles), V = 5 and E = 8, so F + V − E = 5 + 5 − 8 = 2. Euler's formula holds for both.

Describe how the Sierpinski carpet and Koch snowflake are made, and how area and perimeter change at each step.5 marks

Sierpinski carpet: start with a square, divide it into 9 equal smaller squares and remove the middle one. Repeat the same process on each of the 8 remaining squares. At each step the number of squares is multiplied by 8 and each remaining part keeps 8/9 of the area, so after n steps there are 8ⁿ squares and the area left is (8/9)ⁿ of the original; the area keeps decreasing towards zero. Koch snowflake: start with an equilateral triangle. Divide each side into three equal parts, draw an equilateral triangle outward on the middle part and remove its base. Each side is replaced by 4 segments, each one-third as long, so the perimeter is multiplied by 4/3 at every step and keeps growing without limit, while the area grows only a little and stays within a bounded region.

Quick revision

Fractal = self-similar repeating pattern.
Carpet: keep 8 of 9; gasket: keep 3 of 4.
Koch: perimeter grows by 4/3 each step.
F + V − E = 2.

Exam tips

For fractal questions, write the pattern for step 1, 2 and 3 first, then generalise.
Check Euler's formula on a cube (6 + 8 − 12 = 2) before using it on other solids.

Practice MCQs

A cube has how many faces
1. 4
2. 6
3. 8
4. 12
Show answer
Answer: (b) 6. A cube has 6 square faces.
A flat pattern that folds into a solid is called a
1. projection
2. prism
3. fractal
4. net
Show answer
Answer: (d) net. A cube can be made from a net of six squares.
Euler's formula for polyhedra is
1. F + V + E = 2
2. F − V + E = 2
3. F + V − E = 2
4. F × V = E
Show answer
Answer: (c) F + V − E = 2. For a cube 6 + 8 − 12 = 2.
In the Sierpinski carpet, how many small squares remain after the first step
1. 3
2. 9
3. 8
4. 4
Show answer
Answer: (c) 8. The square is cut into 9 and the middle one is removed.
In the Sierpinski gasket, how many triangles remain after 2 steps
1. 6
2. 9
3. 12
4. 16
Show answer
Answer: (b) 9. 3² = 9 triangles.
A triangular prism has 5 faces and 6 vertices. Its number of edges is
1. 8
2. 9
3. 10
4. 11
Show answer
Answer: (b) 9. E = F + V − 2 = 5 + 6 − 2 = 9.
The perimeter of a Koch snowflake starts at 9 cm. After one step it is
1. 9 cm
2. 10 cm
3. 12 cm
4. 27 cm
Show answer
Answer: (c) 12 cm. 9 × 4/3 = 12 cm.
What fraction of the original area remains in the Sierpinski carpet after 2 steps
1. 8/9
2. 16/81
3. 1/9
4. 64/81
Show answer
Answer: (d) 64/81. (8/9)² = 64/81.

All chapters of Class 8 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.