DEMO
Look around freely · sample student, nothing is saved
Get the app
11
NCERT Solutions · Maths Ch 11
Exploring Some Geometric Themes
31 questions · 7 exercises
Closed
Open all
Ex 11.1
Ex 11.2
Ex 11.3
Ex 11.4
Ex 11.5
Ex 11.6
Ex 11.7
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 11.1 · 3 QUESTIONS
Sierpinski carpet and triangle
Q1
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.
(2 marks)
Q2
Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle. (Carpet: squares that remain and holes.)
(3 marks)
Q3
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.
(3 marks)
2
EXERCISE 11.2 · 3 QUESTIONS
Koch snowflake
Q1
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake.
(2 marks)
Q2
Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.
(2 marks)
Q3
Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a side length of 1 unit.
(3 marks)
3
EXERCISE 11.3 · 11 QUESTIONS
Build it in Your Imagination
Q2
Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?
(3 marks)
Q3
Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?
(3 marks)
Q4
Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?
(3 marks)
Q5
Describe a solid and a viewpoint that would result in a profile with a square outline.
(2 marks)
Q6
Describe a solid and a viewpoint that would result in a profile with a circular outline.
(2 marks)
Q7
Describe a solid and a viewpoint that would result in a profile with a triangular outline.
(2 marks)
Q8
A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint.
(2 marks)
Q9
A solid with a circular profile from one viewpoint and a triangular one from another viewpoint.
(2 marks)
Q10
A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint.
(2 marks)
Q11
A solid with a trapezium-shaped profile from one viewpoint and a circular one from another viewpoint.
(2 marks)
Q12
A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint.
(2 marks)
4
EXERCISE 11.4 · 3 QUESTIONS
Nets of solids
Q1
Which of the following are the nets of a cube? (six figures (i)–(vi)) (i) 2 squares in the top row (columns 1–2) and 4 squares in the next row (columns 2–5) (ii) 3 squares in the top row (columns 1–3) and 3 in the next row (columns 3–5) (iii) a staircase of 2, 2, 2 squares (iv) a cross: a row of 4 with one square above the third square and one below the third (v) a row of 4 with a column of 2 squares hanging below the second square (vi) a row of 4 with one square above the third and one below the second
(3 marks)
Q2
A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be obtained from the other by a rotation or a flip. Find all the 11 nets of a cube.
(3 marks)
Q3
Draw a net of a cuboid having side lengths: (i) 5 cm, 3 cm and 1 cm (ii) 6 cm, 3 cm and 2 cm
(3 marks)
5
EXERCISE 11.5 · 3 QUESTIONS
Projections: front, top, side
Q1
Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?
(3 marks)
Q2
Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to the vertical, horizontal and side planes: cube, cuboid, parallelepiped, cylinder, cone, prism, and pyramid.
(5 marks)
Q3
Match each of the following objects with its projections. (objects: a jug, a funnel, a hammer, a car, a ramp with a box, a chair, a ceiling fan, a bucket)
(5 marks)
6
EXERCISE 11.6 · 4 QUESTIONS
Views of combinations of cubes
Q1
Draw the top view, front view and the side view of each of the following combinations of identical cubes. (six figures)
(5 marks)
Q2
Imagine eight identical cubes, glued together along faces to form the letter 'C'. (i) This looks like a 'C' from the front. What does it look like from the side? From the top? (ii) Glue additional cubes to make a shape that looks like 'C' from the front and 'A' from the top. (iii) Now, can you glue even more cubes to make it look like 'C' from the front, 'A' from the top, and 'F' from the side? (iv) Can you think of other letter combinations to make with a single combination of cubes in this manner?
(3 marks)
Q3
Which solid corresponds to the given top view, front view and side view? (Given: front view, top view and side view of a stepped solid; options (i)–(vii).)
(3 marks)
Q4
Using identical cubes, make a solid that gives the following projections. (Three sets of top, front and side views, (i)–(iii), (iv)–(vi), (vii)–(ix).)
(5 marks)
7
EXERCISE 11.7 · 4 QUESTIONS
Isometric drawings
Q1
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?
(3 marks)
Q2
Draw the following figures on the isometric grid. (three solids: an L-shaped bar, a T-shaped solid, a staircase)
(3 marks)
Q3
Is there anything strange about the path of this ball? Recreate it on the isometric grid.
(2 marks)
Q4
Observe this triangle (an impossible 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? (ii) Recreate this on an isometric grid. (iii) Why does the illusion work?
(3 marks)