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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 4: Exploring Some Geometric Themes

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Self-similarity and fractal patterns

From daily life

Look at a fern leaf: each small leaflet looks like the whole leaf. A kolam or a temple tower can repeat a design inside itself again and again. This "copy of itself inside itself" idea is self-similarity.

Simple definition

A fractal is a self-similar shape made by repeating the same rule again and again. Sierpinski carpet: cut a square into 9 equal squares, remove the middle one, repeat on the 8 left. Sierpinski triangle: join the midpoints of a triangle, remove the middle triangle, repeat on the 3 left.

In the figure: Carpet after step 2: 64 small squares.

In the figure: Triangle after step 3: 27 small triangles.

Rule 1: Counting the carpet — Each step multiplies the number of squares by 8: stage n has 8^n squares.

  • Easy: Step 1: 8 squares.
  • Medium: Step 2: 8 × 8 = 64 squares.
  • Exam-type: Step 3: 64 × 8 = 512 squares; each has side 1/27 of the original.

Rule 2: Counting the triangle — Each step multiplies the number of triangles by 3: stage n has 3^n triangles.

  • Easy: Step 1: 3 triangles.
  • Medium: Step 2: 9 triangles; 3 were removed in this step.
  • Exam-type: Step 3: 27 triangles, each with side 1/8 of the original.

Rule 3: Size of the pieces — Carpet pieces shrink to 1/3 of the side each step; triangle pieces shrink to 1/2 of the side.

  • Easy: Carpet side 27 cm: pieces 9 cm, then 3 cm, then 1 cm.
  • Medium: Triangle side 16 cm: pieces 8 cm, then 4 cm.
  • Exam-type: Carpet step 2 square: (1/3) × (1/3) = 1/9 of the side.
Remember

These shapes are built by repeating, so you can find the next stage just by applying the rule once more to every piece.

Common mistake

Counting only the first-step pieces: after step 2 the carpet has 64 squares, not 16, because each of the 8 squares is treated again.

Shortcut

Carpet ×8 (eight left), triangle ×3 (three left). Memory clue: "Carpet-8, Triangle-3".

Spot the difference

PointSierpinski carpetSierpinski triangle
Start witha squarea triangle
Pieces after step n8^n3^n
Side of each piece(1/3)^n(1/2)^n

Practice questions

Easy
  1. 1How many equal squares is a square divided into in the first carpet step?
  2. 2How many squares are left after step 1 of the carpet?
  3. 3How many triangles are left after step 1 of the Sierpinski triangle?
Medium
  1. 4How many squares does the carpet have after step 2?
  2. 5How many triangles are there after step 3 of the Sierpinski triangle?
  3. 6A carpet of side 27 cm: find the side of a piece after step 2.
Challenging
  1. 7How many squares does the carpet have after step 3?
  2. 8How many triangles are removed in step 2 of the Sierpinski triangle?
Show answers (try them yourself first)
  1. 19
  2. 28
  3. 33
  4. 464
  5. 527
  6. 63 cm
  7. 7512
  8. 83

Area in the Sierpinski carpet and triangle

From daily life

A baker cuts a square cake into 9 equal pieces and eats the middle one: 8/9 of the cake is left. Then he does the same on each of the 8 pieces: (8/9) of 8/9 is left.

Simple definition

Each carpet step keeps 8/9 of the area. Each triangle step keeps 3/4 of the area. After n steps the area left = original area × (8/9)^n for the carpet and × (3/4)^n for the triangle. Area removed = 1 − the fraction left.

Rule 1: Area of the carpet — Multiply the area by 8/9 for every step.

  • Easy: Side 9 cm: area 81; step 1 leaves 72 cm².
  • Medium: Side 27 cm: area 729; removed 81; left 648 cm².
  • Exam-type: After 2 steps 64/81 is left; after 3 steps 512/729.

Rule 2: Area of the triangle — Multiply the area by 3/4 for every step.

  • Easy: Area 64 cm²: step 1 leaves 48 cm².
  • Medium: Area 64 cm²: step 2 leaves 36 cm², step 3 leaves 27 cm².
  • Exam-type: Area 144 cm²: after 2 steps 144 × 9/16 = 81 cm².

Rule 3: Area removed — Subtract the fraction left from 1.

  • Easy: Carpet step 1 removes 1/9; triangle step 1 removes 1/4.
  • Medium: Triangle after 2 steps: 1 − 9/16 = 7/16 removed.
  • Exam-type: Carpet after 2 steps: 1 − 64/81 = 17/81 removed.
Remember

The area left shrinks towards zero as the steps go on, even though the shape never disappears completely.

Common mistake

Subtracting 1/9 each time instead of multiplying by 8/9. After 2 steps the carpet keeps 64/81, not 7/9.

Shortcut

Keep fraction, multiply again. Memory clue: "Carpet 8 of 9, triangle 3 of 4 - multiply, don't subtract".

Spot the difference

PointCarpetTriangle
Kept each step8/93/4
After 2 steps64/819/16
Removed after 2 steps17/817/16

Practice questions

Easy
  1. 1What fraction of the carpet area is left after step 1?
  2. 2A carpet has side 9 cm. Find the area left after step 1.
  3. 3A triangle of area 64 cm² is used. Find the area left after step 1.
Medium
  1. 4Find the fraction of the carpet area left after step 2.
  2. 5A triangle of area 64 cm²: find the area left after step 3.
  3. 6Find the fraction of the triangle removed after 2 steps.
Challenging
  1. 7A carpet has side 27 cm. Find the area removed in step 1.
  2. 8Which has more area left after step 2, the carpet or the triangle?
Show answers (try them yourself first)
  1. 18/9
  2. 272 cm²
  3. 348 cm²
  4. 464/81
  5. 527 cm²
  6. 67/16
  7. 781 cm²
  8. 8Carpet (64/81 > 9/16)

The Koch snowflake

From daily life

Imagine a coastline: the closer you look, the more bays and bumps you see, so the coast measured with a smaller ruler gets longer. The Koch snowflake is a neat model of this.

Simple definition

Start with an equilateral triangle. In every step, cut each side into three equal parts, remove the middle part and put a small equilateral "bump" there (two pieces). Each side becomes 4 pieces, each 1/3 as long, so the perimeter is multiplied by 4/3.

31248

In the figure: Number of sides: 3, 12, 48, ...

Rule 1: Count the sides — Multiply the number of sides by 4 every step: 3 × 4^n.

  • Easy: Stage 1: 12 sides.
  • Medium: Stage 2: 48 sides; stage 3: 192 sides.
  • Exam-type: One original side becomes 4 pieces, then 16 pieces.

Rule 2: Length of one piece — Divide by 3 every step: original side ÷ 3^n.

  • Easy: Side 27 cm: stage 1 pieces 9 cm.
  • Medium: Side 27 cm: stage 2 pieces 3 cm; stage 3 pieces 1 cm.
  • Exam-type: Side 81 cm: stage 2 pieces 9 cm.

Rule 3: Perimeter — Start with 3 × side, then multiply by 4/3 for each step.

  • Easy: Side 9 cm: 27 → 36 cm at stage 1.
  • Medium: Side 9 cm: stage 2 is 36 × 4/3 = 48 cm.
  • Exam-type: Side 6 cm: 18 → 24 → 32 cm.
Remember

The perimeter grows without any limit, but the whole snowflake always fits inside a circle, so its area stays finite.

Common mistake

Thinking the new bump is added to just one side, or that each piece becomes 1/2 as long. Every side changes, and each new piece is 1/3 as long.

Shortcut

Sides ×4, length ÷3, perimeter ×4/3. Memory clue: "Four more, a third as long".

Spot the difference

PointSidesPerimeter
Each step× 4× 4/3
Stage 2 (start 3 and 27)4848
Trendgrowsgrows without limit

Practice questions

Easy
  1. 1How many sides does a Koch snowflake have at stage 1?
  2. 2Side 27 cm: find the length of a piece at stage 1.
  3. 3Side 9 cm: find the perimeter at stage 1.
Medium
  1. 4How many sides does it have at stage 3?
  2. 5Side 9 cm: find the perimeter at stage 2.
  3. 6Side 81 cm: find a piece at stage 2.
Challenging
  1. 7Side 6 cm: find the perimeter at stage 2.
  2. 8A segment of length 9 becomes a Koch curve. Find its length at stage 2.
Show answers (try them yourself first)
  1. 112
  2. 29 cm
  3. 336 cm
  4. 4192
  5. 548 cm
  6. 69 cm
  7. 732 cm
  8. 816
Live figure

The Koch snowflake

In the app you can move this figure with your finger.

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Faces, edges, vertices and nets

From daily life

Take a cardboard box. The flat sides are its faces, the lines where two faces meet are its edges, and the pointed corners are its vertices. If you cut along some edges and flatten the box, you get a net.

Simple definition

For any solid with flat faces: V − E + F = 2 (V vertices, E edges, F faces). A prism with an n-sided base has n + 2 faces, 3n edges, 2n vertices. A pyramid with an n-sided base has n + 1 faces, 2n edges, n + 1 vertices.

F = 6E = 12V = 88 − 12 + 6 = 2

In the figure: A cuboid: 8 − 12 + 6 = 2.

Rule 1: Prisms — Count with the base sides n: faces n + 2, edges 3n, vertices 2n.

  • Easy: Pentagonal prism (n = 5): 7 faces, 15 edges, 10 vertices.
  • Medium: Hexagonal prism (n = 6): 8 faces, 18 edges, 12 vertices; 6 are rectangles.
  • Exam-type: Prism with a 10-sided base: 12 faces, 30 edges, 20 vertices.

Rule 2: Pyramids and Euler's rule — Pyramid: n + 1, 2n, n + 1. Check or find a missing count with V − E + F = 2.

  • Easy: Square pyramid (n = 4): 5 faces, 8 edges, 5 vertices.
  • Medium: V = 12, E = 30: F = 2 − 12 + 30 = 20.
  • Exam-type: Tetrahedron: 4 faces, 6 edges, 4 vertices; 4 − 6 + 4 = 2.

Rule 3: Nets — List each face of the solid; the net must show all of them, joined along edges.

6 squares

In the figure: A cube net: six squares joined edge to edge that fold without overlap.

  • Easy: Cube: 6 squares. Triangular prism: 2 triangles + 3 rectangles.
  • Medium: Square pyramid: 1 square + 4 triangles.
  • Exam-type: Cylinder: 2 circles + 1 rectangle.
Remember

A solid with curved surfaces (sphere, cylinder, cone) does not follow n + 2 style counts, but its net still shows its flat and curved parts.

Common mistake

Confusing faces and edges, or forgetting the top and bottom of a prism. A prism with a triangle base has 2 + 3 = 5 faces.

Shortcut

V − E + F = 2. Memory clue: "Very Easy Formula equals Two".

Spot the difference

PointPrism (n-sided base)Pyramid (n-sided base)
Facesn + 2n + 1
Edges3n2n
Vertices2nn + 1

Practice questions

Easy
  1. 1How many faces, edges and vertices does a cuboid have?
  2. 2A prism has a 5-sided base. Find its faces.
  3. 3What does the net of a cube contain?
Medium
  1. 4A pyramid has a 10-sided base. Find its edges.
  2. 5V = 12 and E = 30. Find F.
  3. 6What does the net of a triangular prism contain?
Challenging
  1. 7A prism has a 10-sided base. Find its vertices.
  2. 8A solid has V = 8 and F = 6. Find E.
Show answers (try them yourself first)
  1. 16, 12, 8
  2. 27
  3. 36 squares
  4. 420
  5. 520
  6. 62 triangles, 3 rectangles
  7. 720
  8. 812

Views and stacks of cubes

From daily life

A child builds a tower of blocks. From above you see only the tops; from the front you see how tall each column is. The same solid can look like a circle from one side and a rectangle from another.

Simple definition

The top view, front view and side view are the flat outlines of a solid seen from above, the front and the side. For a stack of cubes, the top view has one square for every column, and the total number of cubes is the sum of the column heights.

213112top view: 6 squarescubes: 2+1+3+1+1+2 = 10

In the figure: Heights in the top view add up to the total number of cubes: 10.

Rule 1: Profiles of simple solids — Imagine the solid passing through a wall: the hole shows the profile.

  • Easy: Cylinder: circle from above, rectangle from the side.
  • Medium: Cone: circle from above, triangle from the side. Sphere: circle from everywhere.
  • Exam-type: A cylinder whose height equals its diameter shows a square from the side and a circle from above.

Rule 2: Counting cubes in a stack — Add the heights of all columns. Top view squares = number of columns.

  • Easy: Heights 3, 2, 1: 6 cubes.
  • Medium: Heights 4, 1, 2, 3: 10 cubes and 4 squares in the top view.
  • Exam-type: Block 3 long, 2 wide, 4 high: 3 × 2 × 4 = 24 cubes.

Rule 3: Front view and special views — The front view shows the tallest column in each position. A cube drawn in isometric style has a hexagonal outline. Sunlight falling straight down gives a shadow of the same shape and size.

  • Easy: Heights 2, 1, 3, 1: tallest in the front view is 3.
  • Medium: Heights 1, 4, 2: the front outline reaches 4.
  • Exam-type: Square pyramid: square from above, triangle from the side.
Remember

One view is never enough: a circle top view could be a sphere, a cylinder or a cone, so always check at least two views.

Common mistake

Counting only the cubes you can see. Hidden cubes behind and below also count in the total.

Shortcut

Top = columns, add heights = cubes. Memory clue: "Top counts towers, heights count blocks".

Spot the difference

PointTop viewFront view
Showsthe columnsthe heights
Cylindercirclerectangle
Conecircletriangle

Practice questions

Easy
  1. 1What does a cylinder show from above?
  2. 2Stacks have heights 3, 2, 1. How many cubes in all?
  3. 3What does a cone show from the side?
Medium
  1. 4Stacks have heights 4, 1, 2, 3. How many cubes in all?
  2. 5For heights 2, 1, 3, 1 how many squares are in the top view?
  3. 6How many unit cubes fill a block 3 long, 2 wide and 4 high?
Challenging
  1. 7Which solid shows a square from the side and a circle from above?
  2. 8Heights 1, 4, 2: find the tallest height in the front view and the total cubes.
Show answers (try them yourself first)
  1. 1circle
  2. 26
  3. 3triangle
  4. 410
  5. 54
  6. 624
  7. 7Cylinder with height equal to diameter
  8. 84 and 7