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Class 8 Ganita Prakash · Part 2 · Chapter 4
Class 8 Ganita Prakash · Part 2 · Chapter 4
Exploring Some Geometric Themes
Page-wise textbook solutions (pages 70–102, Part 2) · Every step · Figures · Common mistakes · Tips
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๐️ Key ideas of this chapter
- Fractals repeat the same pattern at smaller scales: carpet \(8^n\) squares, triangle \(3^n\) triangles, Koch snowflake \(3 \times 4^n\) sides with perimeter \(3(\tfrac{4}{3})^n\).
- Prism with n-gon ends: \(n+2\) faces, \(3n\) edges, \(2n\) vertices. Pyramid: \(n+1\), \(2n\), \(n+1\).
- A cube has 11 nets; a regular tetrahedron has 2; a sphere has none.
- Shortest path on a cuboid = shortest straight line on some net (check all unfoldings).
- Front, top and side views are projections; parallel lines stay parallel. Isometric view of a cube = regular hexagon.
๐ Textbook Page 71
๐ Page 71Sierpinski Carpet
Draw up to Step 2. Find the number of remaining squares Rโ and holes Hโ at step n.
1Each remaining square becomes 8 squares: \(R_{n+1} = 8R_n\) → \(R_n = 8^n\).
2Each remaining square makes one new hole: \(H_{n+1} = H_n + R_n\).
3\(H_n = 1 + 8 + 8^2 + \cdots + 8^{n-1} = \dfrac{8^n - 1}{7}\)
4Check: H₁ = 1, H₂ = 9, H₃ = 73.
✅ Answer: \(R_n = 8^n\), \(H_n = \dfrac{8^n-1}{7}\)
๐ Textbook Page 72
๐ Page 72Show
Show that joining the midpoints of an equilateral triangle gives 4 identical equilateral triangles.
1A corner triangle has two sides of length a/2 (half-sides) with a 60° angle between them.
2Isosceles with a 60° apex → the other two angles are (180° − 60°) ÷ 2 = 60° each → equilateral, so the third side is a/2.
3So all three middle segments are a/2: the central triangle is also equilateral with side a/2.
✅ Answer: All 4 triangles are equilateral with side a/2.
๐ Page 72Figure it Out · Q1
Draw the first steps (at least Step 2) of the Sierpinski Triangle.
1Step 1: join midpoints, remove the middle triangle.
2Step 2: repeat on each of the 3 remaining triangles.
✅ Answer: See the figure.
๐ Page 72Figure it Out · Q2
Number of holes and remaining triangles at each step of the Sierpinski Triangle.
1Each remaining triangle gives 3: \(T_n = 3^n\).
2Each remaining triangle makes one new hole: \(H_{n+1} = H_n + T_n\).
3\(H_n = 1 + 3 + \cdots + 3^{n-1} = \dfrac{3^n - 1}{2}\): 0, 1, 4, 13, 40 …
✅ Answer: Triangles \(3^n\); holes \(\dfrac{3^n-1}{2}\)
๐ Page 72Figure it Out · Q3
Area remaining at step n (starting area 1) for the carpet and the triangle.
1Carpet: each step keeps 8 of 9 parts → multiply by \(\tfrac{8}{9}\).
2Triangle: each step keeps 3 of 4 parts → multiply by \(\tfrac{3}{4}\).
✅ Answer: Carpet \((\tfrac{8}{9})^n\); triangle \((\tfrac{3}{4})^n\) — both get closer and closer to 0!
๐ Textbook Page 73
๐ Page 73Figure it Out · Q1
Draw the Koch Snowflake up to Step 2.
1Divide each side into 3, raise an equilateral 'bump' on the middle third, remove the base of the bump.
2Repeat on every new side.
✅ Answer: See the figure.
๐ Page 73Figure it Out · Q2
Number of sides at step n of the Koch Snowflake.
1Each side becomes 4 sides.
23 → 12 → 48 → 192 …
✅ Answer: \(3 \times 4^n\)
๐ Page 73Figure it Out · Q3
Perimeter at step n (start side 1).
1Sides: \(3 \times 4^n\); each side length \((\tfrac{1}{3})^n\).
2Perimeter = \(3 \times 4^n \times (\tfrac{1}{3})^n = 3 \times (\tfrac{4}{3})^n\)
✅ Answer: \(3(\tfrac{4}{3})^n\) — it grows without limit, though the area stays finite!
๐ Textbook Page 76
๐ Page 76Imagine · Q2
Cut off the four corners of a square, each cut joining midpoints of adjacent sides. What is left? Can the 4 corners make another square?
1The cuts join the 4 midpoints → a tilted square (half the area).
2Each corner is a right isosceles triangle; put the 4 right angles together at a point → another square of the same size.
✅ Answer: A tilted square of half the area; the 4 corners make an equal square.
๐ Page 76Imagine · Q3
Mark each side of an equilateral triangle into thirds and cut off the corners up to the marks. What shape remains?
1Each corner cut removes an equilateral triangle of side a/3.
26 sides remain, each a/3, all angles 120°.
✅ Answer: A regular hexagon.
๐ Page 76Imagine · Q4
Mark the sides of a square into thirds and cut off the corners up to the marks. What is left?
14 sides of length a/3 remain from the original sides.
24 new slanted sides of length \(\tfrac{a\sqrt{2}}{3} \approx 0.47a\).
3All 8 angles are 135°.
✅ Answer: An octagon with equal angles but NOT all sides equal (not regular).
⚠️ Common mistake: Calling it a regular octagon — the slanted sides are longer.
๐ Textbook Page 77
๐ Page 77Profiles · Q5–Q7
Name a solid and viewpoint giving a (5) square (6) circular (7) triangular outline.
1(5) Cube from any face; a cylinder (height = diameter) from the side.
2(6) Sphere from anywhere; cylinder or cone from the top.
3(7) Cone or square pyramid from the side; triangular prism from its end.
✅ Answer: Many answers are possible.
๐ Page 77Profiles · Q8–Q12
Find solids with: (8) rectangle & circle (9) circle & triangle (10) rectangle & triangle (11) trapezium & circle (12) pentagon & rectangle profiles.
1(8) Cylinder: side → rectangle, top → circle.
2(9) Cone: top → circle, side → triangle.
3(10) Triangular prism: side → rectangle, end → triangle.
4(11) Bucket (frustum of a cone): side → trapezium, top → circle.
5(12) Pentagonal prism: end → pentagon, side → rectangle.
✅ Answer: Cylinder, cone, triangular prism, bucket (frustum), pentagonal prism. The solids are not unique (e.g. a tent shape also gives rectangle & triangle).
๐ Textbook Page 79
๐ Page 79Think
A prism has 10-sided end faces. Faces, edges, vertices? n-sided? A pyramid with a 10-sided base? n-sided?
1Prism: 2 end faces + n side faces = n + 2; edges: n + n + n = 3n; vertices: 2n.
2Pyramid: base + n triangles = n + 1; edges: n (base) + n (slant) = 2n; vertices: n + 1.
3Check Euler: F + V − E = 2 in both cases.
✅ Answer:
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Prism, n = 10 | 12 | 30 | 20 |
| Prism, n | n + 2 | 3n | 2n |
| Pyramid, n = 10 | 11 | 20 | 11 |
| Pyramid, n | n + 1 | 2n | n + 1 |
๐ Textbook Page 80
๐ Page 80Figure it Out · Q1
Which of (i)–(vi) are nets of a cube?
1Imagine folding: a row of 4 squares wraps around as 4 side faces; one square must go on top and one at the bottom, on OPPOSITE sides of the row.
2(i) The two extra squares both end up on the same side → overlap. ✘
3(v) The squares hanging below cover the same face twice. ✘
4(ii), (iii), (iv), (vi) fold up correctly. ✔
✅ Answer: (ii), (iii), (iv) and (vi) are nets; (i) and (v) are not.
๐ Textbook Page 81
๐ Page 81Figure it Out · Q2
Find all 11 nets of a cube.
1Six nets have a row of 4 with one square above and one below (1-4-1).
2Three have the pattern 2-3-1, one is 2-2-2 (staircase), one is 3-3.
✅ Answer: See the 11 nets.
๐ Page 81Figure it Out · Q3
Draw a net of a cuboid (i) 5 × 3 × 1 cm (ii) 6 × 3 × 2 cm.
1A cuboid has 3 pairs of equal rectangles: l × w, l × h, w × h.
2(i) two each of 5 × 3, 5 × 1, 3 × 1.
3(ii) two each of 6 × 3, 6 × 2, 3 × 2.
4Arrange 4 of them in a band (l × w, l × h, l × w, l × h) and put the two w × h ends on either side.
✅ Answer: See the nets (draw to scale in cm).
๐ Page 81Think
Which drawings are nets of a regular tetrahedron? Draw one with measurements. Also a square pyramid.
1Net 1: a big triangle divided into 4 (fold the 3 corners up).
2Net 2: a strip of 4 triangles.
3The other two shapes (a row of 3 with one hanging) don't close up — two faces overlap.
4Square pyramid: a square of side 5 cm with 4 equal isosceles triangles on its sides.
✅ Answer: 1st and 3rd are nets; only 2 nets exist.
๐ Textbook Page 82
๐ Page 82Think
What are the sides of the rectangle in a cylinder's net? What is the net of a cone?
1The rectangle wraps around the curved surface: one side is the height h.
2The other side goes once around the circle: \(2\pi r\) (circumference).
3A cone unrolls into a sector of a circle with centre O (the apex) and radius = slant height l, plus a circle for the base.
✅ Answer: Rectangle h × 2ฯr; cone → sector + circle.
๐ Page 82Math Talk
What surface do you get from a sector-shaped net if O is NOT the centre of the curved edge?
1The points on the edge are at different distances from the apex.
2When rolled up, the rim is not at one level — it is uneven/slanted.
✅ Answer: A cone-like surface whose rim is not flat (it wobbles and won't stand evenly).
๐ Page 82Make
Draw a net for a triangular prism and an octahedron.
1Triangular prism: 3 rectangles in a row (e.g. 8 cm × 4 cm each) with an equilateral triangle (side 4 cm) on each end of the middle rectangle.
2Octahedron: 8 equilateral triangles (as in the book's net), e.g. side 5 cm; fold so 4 meet at each vertex.
✅ Answer: See the description.
๐ Textbook Page 84
๐ Page 84Net of a sphere?
Can paper wrap a ball with no wrinkles, gaps or overlaps?
1Paper can bend in one direction only (like a cylinder or cone).
2A sphere curves in every direction, so flat paper must wrinkle or be cut into many pieces (like a football's panels).
✅ Answer: No — a sphere has no net.
๐ Page 84Shortest path
How do we find and check the shortest path for the ant on the surface of a cuboid?
1Unfold the cuboid into a net so that the ant and the laddu are on a flat surface.
2On a flat net the shortest path is a straight line — but the line must stay inside the net.
3Try different unfoldings; the shortest straight line among them is the answer.
✅ Answer: Unfold, draw a straight line, compare unfoldings.
๐ก Tip: A path that bends on the net (like the blue one) is never the shortest.
๐ Textbook Page 87
๐ Page 87Try This
Room 30 × 12 × 12 cm: the ant (1 cm from the top, middle of one end) and the laddu (1 cm from the bottom, middle of the other end). Shortest path?
1Straight across the top: 1 + 30 + 11 = 42 cm.
2Clever unfolding (over the top, a side and the floor): right triangle with legs 24 and 32.
3\(d^2 = 24^2 + 32^2 = 576 + 1024 = 1600\) → d = 40
✅ Answer: 40 cm — shorter than the obvious 42 cm path!
๐ Textbook Page 89
๐ Page 89Projection of a line
Compare a line's length l with its projection p. When are they equal?
1AECD is a rectangle, so AE = p; triangle AEB is right-angled at E.
2AB = l is the hypotenuse → l ≥ p.
✅ Answer: p ≤ l; p = l only when the line is parallel to the plane.
๐ Page 89Math Talk
What can the projection of a square, a parallelogram and a regular n-gon look like?
1Parallel lines project to parallel lines (or to one line).
2Square → square, rectangle, parallelogram, or a line segment (edge-on).
3Parallelogram → always a parallelogram (or segment) — never another quadrilateral.
4Regular n-gon → an n-gon with opposite sides still parallel (if n even), squashed in one direction, or a segment.
✅ Answer: Parallelism is kept; lengths and angles may change.
๐ Page 89Think
How do the projections of a cube and a cone look? Find another object with the same projection as a cone.
1Cube: square (face-on), rectangle (tilted one way), hexagon (tilted both ways).
2Cone: triangle from the side, circle from the top.
3A flat triangular card or a square pyramid also gives a triangle; a disc gives a circle.
✅ Answer: Projections don't identify an object — so we use 3 views.
๐ Textbook Page 92
๐ Page 92Figure it Out · Q1
Relation between the lengths of the front, top and side views of a line?
1If the line goes a across, b deep and c up: front = \(\sqrt{a^2+c^2}\), top = \(\sqrt{a^2+b^2}\), side = \(\sqrt{b^2+c^2}\).
2Each view is at most the true length \(\sqrt{a^2+b^2+c^2}\).
3Sum of the squares of the three views = 2 × (true length)².
✅ Answer: Each view ≤ true length; front² + top² + side² = 2l².
๐ Page 92Figure it Out · Q2
Front, top and side views of standard solids (standing upright).
1Cube: square, square, square.
2Cuboid: rectangles (of different sizes).
3Parallelepiped: parallelogram/rectangles depending on the slant.
4Cylinder: rectangle, circle, rectangle.
5Cone: triangle, circle with centre dot, triangle.
6Triangular prism (lying): rectangle, rectangle, triangle.
7Square pyramid: triangle, square with diagonals, triangle.
✅ Answer:
| Solid | Front | Top | Side |
|---|---|---|---|
| Cube | square | square | square |
| Cylinder | rectangle | circle | rectangle |
| Cone | triangle | circle + dot | triangle |
| Square pyramid | triangle | square + diagonals | triangle |
| Triangular prism (lying) | rectangle | rectangle | triangle |
๐ Textbook Page 93
๐ Page 93Figure it Out · Q3
Match each object with its front, top and side projections (rows of the table).
1Look for a special shape in each row: wheels → car; a circle with a spout → mug; three blades → fan.
2Row 1: car · Row 2: tiffin/container · Row 3: slide · Row 4: chair · Row 5: ceiling fan · Row 6: funnel · Row 7: hammer · Row 8: mug
✅ Answer: Mug → row 8, Funnel → row 6, Hammer → row 7, Car → row 1, Slide → row 3, Chair → row 4, Fan → row 5, Container → row 2
๐ Textbook Page 94
๐ Page 94Shadows
What happens to a shadow as the torch moves closer to or farther from the object? Why?
1Light from a torch spreads out from one point.
2Closer torch → rays spread more → bigger shadow (similar triangles).
3Very far torch (like the Sun) → rays almost parallel → shadow ≈ the projection.
✅ Answer: Closer = bigger shadow; far away = shadow becomes the true projection.
๐ Textbook Page 95
๐ Page 95Figure it Out · Q1
Draw the top, front and side views of each combination of cubes.
1Front view: look along the 'Front' arrow — draw each column of cubes you see, as tall as the tallest cube in it.
2Top view: look down — draw one square for every place where at least one cube stands.
3Side view: look along the 'Side' arrow — draw each column as tall as its tallest cube.
4Check: the width of the front view = width of the top view; the depth of the top view = width of the side view; heights of front and side views match.
✅ Answer: Draw on squared paper using the 3 rules above (one square = one cube face).
๐ก Tip: Build each shape with real cubes or dice and look from the three directions.
๐ Page 95Figure it Out · Q2
8 cubes form a 'C' (3 wide, 4 tall). (i) Side and top views? (ii) Add cubes to make 'C' from the front and 'A' from the top. (iii) Also 'F' from the side? (iv) Other letters?
1(i) Side: a straight column 4 cubes tall (like 'I'). Top: a straight row of 3.
2(ii) Make the top layer and the bottom layer of the C into 'A' shapes (10 cubes each), keeping the C's two middle cubes: total 22 cubes. Front still C, top A.
3(iii) The depth is now 4 (from the A), so the side view must be 4 cubes wide. Keep only cubes allowed by all three letters: with an F that is 4 wide (top row 4, middle rows shorter) it works with about 17 cubes. With an F only 3 wide it is impossible, because the A makes the solid 4 deep.
4(iv) Many others: e.g. 'L' front, 'L' top, 'I' side; 'T' front and 'T' side.
✅ Answer: (i) Side 'I' (4 tall), top a row of 3 (ii) A-shaped top and bottom layers (iii) possible only if the F is 4 wide
๐ Textbook Page 96
๐ Page 96Figure it Out · Q3
Which solid (i)–(vii) has the given front, top and side views?
1Side view: only the BACK part is tall (3 levels); the rest is 1 level high.
2Front view: the tall back part has a step — left higher than right.
3Top view: the low part is an L (full width in the middle, only the left at the front) — no gap/notch.
4Only (ii) has a stepped tall back AND an L-shaped low front without a notch.
✅ Answer: (ii)
๐ก Tip: Rule out options quickly: (i), (iii), (v), (vi) have a notch in front; (iv) and (vii) have no step at the back.
๐ Page 96Figure it Out · Q4
Build a solid with identical cubes for each set of views: (i)–(iii), (iv)–(vi), (vii)–(ix).
1Write the height of each stack on the top view (a 'height map').
2Set 1: 3 × 3 top view with a notch in front-middle; back row has stacks of 2 in the middle and right; all others 1 → 9 cubes.
3Set 2: 2 wide, 3 deep; back row stacks of 2, middle row 1 and 1, front-left 1 → 7 cubes.
4Set 3: back row stacks 2 (left) and 3 (right), middle row 1 and 1, front-left 1 → 8 cubes.
✅ Answer: Set 1: 9 cubes · Set 2: 7 cubes · Set 3: 8 cubes (see the height maps)
๐ Textbook Page 97
๐ Page 97Figure it Out · Q5
Find the number of cubes in the stack.
1We see 1, 2, 3, 4 cubes in the rows, but each higher cube must sit on cubes hidden behind.
2It is a corner stack: layer k is a triangle of 1, 3, 6, 10 cubes.
31 + 3 + 6 + 10 = 20
✅ Answer: 20 cubes (only 10 are visible)
⚠️ Common mistake: Counting only the visible cubes (10).
๐ Page 97Figure it Out · Q6
What shapes can the projection of a cube make?
1Face-on: square.
2Turned about one axis: rectangle (made of 2 visible faces).
3Turned about two axes: hexagon — a regular hexagon when balanced on a corner (isometric).
✅ Answer: Square, rectangle, hexagon.
๐ Page 97Isometric
Balance a cube on a corner. Why do all projected edges have equal length?
1The vertical line through the corner is the cube's long diagonal.
2The 3 edges at the top corner make equal angles with this diagonal (symmetry), so they are shortened by the same amount; the same holds for all 12 edges.
✅ Answer: Equal tilt → equal shortening → a regular hexagon.
๐ Textbook Page 100
๐ Page 100Figure it Out · Q1
Besides the 5 flat Tetris shapes, are there other ways of gluing 4 cubes face-to-face?
1Allow cubes to go 'up' as well as sideways.
2There are 3 more: two 'screw' shapes (mirror images) and a 'tripod' (3 cubes around one corner cube).
✅ Answer: Yes, 3 more → 8 in total.
๐ Textbook Page 101
๐ Page 101Figure it Out · Q2
Draw the given figures on an isometric grid.
1Use | for height, / for depth, \ for length (one grid edge = one unit).
2Draw the base outline first, then the vertical edges, then the top outline.
3Hide lines that are behind; shade one family of faces to make it look solid.
✅ Answer: Follow the grid directions edge by edge.
๐ Page 101Figure it Out · Q3
Is there anything strange about the path of the ball?
1Each step of the path seems to go downhill (or always uphill).
2Yet the path returns to where it started!
3Small parts are possible, but the whole loop cannot exist in 3-D — isometric drawing loses depth, so far and near points can be drawn at the same place.
✅ Answer: It is an impossible staircase (Penrose stairs) — an optical illusion.
๐ Page 101Figure it Out · Q4
The impossible (Penrose) triangle: can it be built from cubes? Why does the illusion work?
1Each corner looks like a real right-angled joint, but the three bars cannot all meet in 3-D.
2Real cubes give it a gap or overlap: a model only looks right from ONE special viewpoint.
3In isometric drawing, a point near you and a point far away along the viewing direction land on the same spot — the drawing joins them.
✅ Answer: No (only from one special angle). The illusion works because isometric pictures lose depth information.
๐ Where do we use this?
- Packaging boxes (nets), engineering and architecture drawings (views, isometric), computer graphics, temple architecture and art (fractals).
๐ Link to higher classes
- Class 9–10: surface area and volume, coordinate geometry in 3-D. Later: geometric series (fractal areas), engineering graphics.
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