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Exploring Some Geometric Themes: questions with answers

Class 8 Maths, Chapter 11 (NCERT). 15 questions by concept, 15 with pictures. Open “Show answer” for the answer and why.

How to teach this chapter · Common misconceptions & worked visual for this chapter

Area remaining in the Sierpinski fractals

Question 1

Each step of the Sierpinski carpet removes the central square of every remaining square, so the area remaining falls to 8/9 of what it was. Each step of the Koch snowflake removes the middle part of every side and raises a bump over it. What happens to the perimeter at each step?

The stated side-replacement ruleOne side is divided into three equal parts. Its middle part is replaced by the two sides of an outward equilateral bump. Matching single marks on the original parts and replacement pieces indicate equal lengths. QUESTION DIAGRAM The stated side-replacement rule Matching marks show equal lengths. Before replacement The three original parts have equal lengths. After the stated replacement Dashed line: the part removed from the old side. Equilateral bump
  1. (a)falls to 8/9 of what it was
  2. (b)stays the same
  3. (c)falls to 2/3 of what it was
  4. (d)rises to 4/3 of what it was
Show answer

(d) rises to 4/3 of what it was

rises to 4/3 of what it was

Counting the squares and holes of the Sierpinski carpet

Question 2

The carpet at step 2 is shown. The cut-and-remove is done once more on every remaining square. How many holes are there after this step?

The Sierpinski carpet at step 2Outlined tiles remain; blank spaces are holes
  1. (a)73
  2. (b)128
  3. (c)25
  4. (d)512
Show answer

(a) 73

Every remaining square gives one new hole, so 82=648^2 = 64 new holes appear, and the 9 old holes stay. That makes 9+64=739 + 64 = 73 holes.

Counting the triangles and holes of the Sierpinski gasket

Question 3

The gasket at step 2 is shown. The step is done once more on every remaining triangle. How many holes are there after this step?

The Sierpinski gasket at step 2Upright tiles remain; inverted gaps are holes
  1. (a)4
  2. (b)26
  3. (c)27
  4. (d)13
Show answer

(d) 13

Every remaining triangle gives one new hole, so 32=93^2 = 9 new holes appear, and the 4 old holes stay. That makes 4+9=134 + 9 = 13 holes.

Fractals in nature and human-made art

Question 4

A print uses a repeating lizard motif to explore tiling. Which feature would also illustrate the self-similarity associated with a fractal?

Lizard motifs on interlocking tilesFour interlocking tiles carry repeated original lizard motifs. Each lizard has a head with two eyes, a body, four bent limbs and a tapering tail. The motifs repeat across the pattern. QUESTION DIAGRAM Lizard motifs on interlocking tiles Original illustrative pattern
  1. (a)the lizards appear at smaller and smaller scales
  2. (b)the lizards are drawn in two colours
  3. (c)the lizards fit together without gaps
  4. (d)the lizards are copied from nature
Show answer

(a) the lizards appear at smaller and smaller scales

the lizards appear at smaller and smaller scales

Changing cube projection outlines

Question 5

A cube is projected orthographically onto a vertical plane in three views: face-on, after a slight turn about a vertical edge, and along a body diagonal in isometric orientation. What are the outlines, in that order?

The three orthographic viewing directionsThree reference cube sketches. Orange arrows indicate, in order, a face-on view, a slight turn about a vertical edge, and a view along the marked body diagonal. Dashed grey edges are hidden in each reference sketch. QUESTION DIAGRAM The three orthographic viewing directions Reference sketches · orange arrows show the viewing directions Face-on Slight turn Along a body diagonal Orange arrows show the viewing direction for each sketch
  1. (a)Square, square, square
  2. (b)Square, non-square rectangle, regular hexagon
  3. (c)Square, circle, triangle
  4. (d)Circle, triangle, pentagon
Show answer

(b) Square, non-square rectangle, regular hexagon

Square, non-square rectangle, regular hexagon

Different objects can share one projection

Question 6

Compare the first and third cuboid rows. Which views reveal their different depths?

Three cuboid rows have equal square front views. Row one has square top and side views; later rows have progressively taller top rectangles and longer side rectangles.FrontTopSide123Views use a common scale
  1. (a)No views; a matching front view fixes the whole cuboid
  2. (b)Front and top views, while the side views agree
  3. (c)Top and side views, while the front views agree
  4. (d)Only the front view
Show answer

(c) Top and side views, while the front views agree

The front squares are equal. The taller top rectangle and longer side rectangle in the third row reveal the greater depth.

Identify basic solids

Question 7

A has parallelogram faces that are not all rectangles. B has only rectangular faces. Which names correctly identify A and B?

A slanted box labelled A and a rectangular box labelled B, with hidden edges dashed.AB
  1. (a)A: triangular prism; B: cone
  2. (b)A: parallelepiped; B: cuboid
  3. (c)A: cuboid; B: triangular pyramid
  4. (d)A: cylinder; B: cuboid
Show answer

(b) A: parallelepiped; B: cuboid

A is a slanted parallelepiped, with flat parallelogram faces. B is a cuboid with rectangular faces. A cylinder would have a curved surface.

Profiles of solids

Question 8

Which pair of different solids can both give the circular outline from suitable viewpoints?

Required profile outlinesRequired outline
  1. (a)A sphere, but no other solid
  2. (b)A cone viewed directly from the side and a face-on cube
  3. (c)A sphere and a cylinder viewed straight at a circular end
  4. (d)A cube viewed straight at a face and a sphere
Show answer

(c) A sphere and a cylinder viewed straight at a circular end

A sphere gives a circular outline from every direction. A cylinder gives a circular outline when viewed straight at either circular end.

Reconstructing solids and hidden edges

Question 9

What do the two dashed vertical lines in the side view represent?

Source-faithful front, top and side projections at equal scale, including hidden edges shown dashed. One compatible opaque unit-cube model is shown below.FrontTopSideAll views use the same cube-edge scaleA model that gives these projectionsFront: lower left Side: lower rightTop: downward
  1. (a)The top viewing direction
  2. (b)Edges hidden by nearer parts of the solid
  3. (c)Two extra columns outside the solid
  4. (d)Two open gaps through the nearer side
Show answer

(b) Edges hidden by nearer parts of the solid

The lines mark edges behind the nearer side. They do not remove cubes or cut gaps through that side.

Tracing a boundary after corner cuts

Question 10

A square has every corner cut off at the adjacent third marks. How many boundary edges come from original sides, and how many come from cuts?

Square with each corner cut between adjacent third marksDashed: original | Solid: remaining
  1. (a)4 original-side parts and 4 cut edges
  2. (b)4 original sides and 0 cut edges
  3. (c)0 original-side parts and 4 cut edges
  4. (d)8 original-side parts and 4 cut edges
Show answer

(a) 4 original-side parts and 4 cut edges

Each original side keeps its middle third, giving four edges. Each of the four corner cuts adds another boundary edge.

Unroll curved surfaces

Question 11

The wrapping side of a cylinder net was labelled with the base radius. What label correctly replaces it?

Cylinder net: a rectangle and two equal circles touching opposite long sides. The horizontal span is w and the upright span is h.wh
  1. (a)Cylinder height
  2. (b)Base diameter
  3. (c)Twice the base circumference
  4. (d)Base circumference
Show answer

(d) Base circumference

The wrapping side must cover the whole circular rim. Neither the radius nor the diameter gives that length.

Counting the sides and perimeter of the Koch snowflake

Question 12

The Koch snowflake shape at step 2 is shown; the starting triangle had sides of 1 unit. What is the perimeter of the shape at step 2?

The Koch snowflake shape at step 2
  1. (a)3 units
  2. (b)16/3 units
  3. (c)4 units
  4. (d)48 units
Show answer

(b) 16/3 units

Every step replaces each side by 4 pieces of one third of its length, so the perimeter is multiplied by 43\frac{4}{3}. From 3 units: 3×43×43=1633 \times \frac{4}{3} \times \frac{4}{3} = \frac{16}{3} units.

Counts from polygonal bases

Question 13

A prism has two pentagonal bases. How many edges does it have?

Transparent prism with two 5-sided polygonal bases and their corresponding vertices joined. All edges, including hidden edges, are shown.5 sides on each base; all edges shown
  1. (a)15
  2. (b)7
  3. (c)5
  4. (d)10
Show answer

(a) 15

The two bases contribute five edges each. Five more edges join corresponding vertices, giving 15 edges.

Cube nets

Question 14

Two extra squares lie on opposite sides of the same square in the row of four. What can be concluded about folding the shown six squares into a cube?

Square arrangements for foldingArrangement D
  1. (a)A cube net must have all six squares in one straight row
  2. (b)Every connected arrangement of six squares folds into a cube
  3. (c)A cube net must have seven square faces
  4. (d)This arrangement folds into a cube without overlapping faces
Show answer

(d) This arrangement folds into a cube without overlapping faces

Folding along the shared edges places the six squares on six different faces. No two squares cover the same face.

Dimensioned cuboid nets

Question 15

What is the total horizontal width of the flat net, from the left edge of D to the right edge of E?

Dimensioned rectangular net with faces A to FFBACDE5 cm3 cm1 cm
  1. (a)7 cm
  2. (b)9 cm
  3. (c)10 cm
  4. (d)5 cm
Show answer

(a) 7 cm

Across D, A and E, the widths are 1 cm, 5 cm and 1 cm. Their total is 7 cm.

Common misconception in Exploring Some Geometric Themes

Open full misconception guide →

Can a boundary grow when its individual segments get shorter?

Wrong

A Koch-curve step shortens the boundary because every replacement segment is shorter.

Why it feels right. I track the length of each piece and overlook the larger count.

Right

Total length depends on the length and the number of segments.

Worked example

The first line spans three units. The replacement path contains four segments of one unit each.
  1. Replace one 3-unit segment with four 1-unit segments.
  2. Their combined length is 4 units.
  3. The shorter pieces make a longer boundary.