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NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2

EXERCISE 10.2 
Question 1. Look at the given map of a city.
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
Answer the following.
(a) Colour the map as follows: Blue-water, red-fire station, orange-library, yellow-schools, Green-park, Pink-College, Purple-Hospital, Brown-Cemetery.
(b) Mark a green ‘X’ at the intersection of Road ‘C’ and Nehru Road, Green ‘Y’ at the intersection of Gandhi Road and Road A.
(c) In red, draw a short street route from library to the bus depot.
(d) Which is further east, the city park or the market?
(e) Which is further south, the primary school or the Sr. Secondary School?

Solution: The shaded (coloured) map according to the required directions is given in part (a). Here various colours are as under
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
(b) Do as directed.
(c) Do as directed.
(d) City park
(e) Senior Secondary school

Question 2. Draw a map of your class using proper scale and symbols for different objects.
Solution: An activity, do it yourself.

Question 3. Draw a map of your school compound using proper scale and symbols for various features like play ground main building, garden, etc.
Solution: It is an activity. Please do it yourself.

Question 4. Draw a map giving instructions to your friend so that she reaches your house without any difficulty.
Solution:It is an activity. Please do it yourself.

FACES, EDGES AND VERTICES
Note: I. A polygon is a 2-D shape made of line segments only.
II. A polyhedron is a 3-D shape made of plane faces.
III. Plural of polyhedron is polyhedra.
IV. A cube, cuboid, pyramid, prism, etc., are polyhedra whereas spheres, cones and cylinders are not polyhydra.

A cuboid has 6 faces, 12 edges and 8 vertices.

Regular Polyhedron: A convex polyhedron is said to be regular if its faces are made up of regular polygons and the same number faces meet at each vertex.

Note: Regular polyhedra are also called platonic solids.

Prism: Prisms are polyhedra whose top and base are congruent polygons and the other faces are parallelograms.

Pyramids: Pyramids are polyhedra with a polygon at its base and all other faces as triangles meeting in a common vertex.
Note: The prisms and pyramids are named after their bases.

EULER’S FORMULA
It is a relation between the faces edges and vertices of a polyhedra. It is given by
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
where
F → Number of faces
V → Number of vertices
E → Number of edges.

SOLUTION TO THINK, DISCUSS AND WRITE
Question: What happens to F, V and E if some parts are sliced off from a solid? (To start with, you may take a plasticine cube, cut a corner off and investigate).
Solution: Consider the following cuboid ABCDEFGH:
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
Case I: The given cuboid has:
(i) Number of faces (F) = 6
(ii) Number of vertices (V) = 8
(iii) Number of edges (E) = 12

We have, F + V = 6 + 8 = 14
          F + V – E = 14 – 12 = 2
i.e.     F + V – E = 2
or the Euler’s formula is verified.

Case II: [A part is sliced off]
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
We have:
Number of faces (F) = 7
Number of vertices (V) = 10
Number of Edges (E) = 15
Also                        F + V = 7 + 10 = 17
and                   F + V – E = 17 – 15 = 2
Here also, Euler’s formula holds true.

EXERCISE 10.3
Question 1. Can a polyhedron have for its faces
(i) 3 triangles?
(ii) 4 triangles?
(iii) a square and four triangles?
Solution: A polyhedron is bounded by four or more than four polygonal faces.
∴ (i) No, it is not possible that a polyhedron has 3 triangles for its faces.
(ii) Yes, 4 triangles can be the faces of a polyhedron.
(iii) Yes, a square and 4 triangles can be the faces of a polyhedron.

Question 2. Is it possible to have a polyhedron with any given number of faces? Hint: Think of a pyramid.
Solution: Yes, it can be possible only if the number of faces is four or more than four.

Question 3. Which are prisms among the following?
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
Solution: Since, a prism is a polyhedron having two of its faces congruent and parallel, where as other faces are parallelogram.
∴ (i) No, a nail is not a prism.
(ii) Yes, unsharpened pencil is a prism.
(iii) No, table weight is not a prism.
(iv) Yes, box is a prism.

Question 4. (i) How are prisms and cylinders alike?
(ii) How are pyramids and cones alike?
Solution: (i) Both of the prisms and cylinders have their base and top as congruent faces and parallel to each other. Also, a prism becomes a cylinder as the number of sides of its base becomes larger and larger.
(ii) The pyramid and cones are alike becomes their lateral faces meet at a vertex. Also a pyramid becomes a cone as the number of sides of its base becomes larger and larger.

Question 5. Is a square prism same as a cube? Explain.
Solution: No, not always, because it can be a cuboid also.

Question 6. Verify Euler’s formula for these solids.
: NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
Solution: (i) In figure (i), we have
                    F = 7, V = 10 and E = 15
∴           F + V = 7 + 10 = 17
       F + V – E = 17 – 15 = 2
i.e.  F + V – E = 2
Thus, Euler’s formula is verified.

(ii) In figure (ii), we have
                     F = 9, V = 9 and E = 16
∴             F + V = 9 + 9 = 18
and   F + V – E = 18 – 16 = 2
i.e.    F + V – E = 2
Thus, Euler’s formula is verified.

Question 7. Using Euler’s formula find the unknown.
NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2
Solution: (i) Here, V = 6 and E = 12
Since                      F + V – E = 2
∴                            F + 6 – 12 = 2
or                                   F – 6 = 2
or                                         F = 2 + 6 = 8

(ii) Here, F = 5 and E = 9
Since                         F + V – E = 2
or                               5 + V – 9 = 2
or                                     V – 4 = 2
or                                           v = 2 + 4 = 6

(iii) Here F = 20 V = 12
Since,                   F + V – E = 2
∴                       20 + 12 – E = 2
or                     20 + 12 –  E = 2
                                32 –  E = 2
or                                     E = 32 – 2 = 30


Question 8. Can polyhedron have 10 faces, 20 edges and 15 vertices?
Solution: Here, F = 10, E = 20 and V = 15
We have:
        F + V – E = 2
∴ 10 + 15 – 20 = 2
       or 25 – 20 = 2
or                  5 = 2 which is not true
i.e.   F + V – E ≠ 2
Thus, such a polyhedron is not possible.

The document NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2 is a part of the Class 8 Course NCERT Textbooks & Solutions for Class 8.
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FAQs on NCERT Solutions for Class 8 Maths - Visualising Solid Shapes - 2

1. What are solid shapes?
Ans. Solid shapes are three-dimensional objects that have length, width, and height. They occupy space and have a definite shape that can be seen and touched. Examples of solid shapes include cubes, cylinders, spheres, and pyramids.
2. How can we visualize solid shapes?
Ans. To visualize solid shapes, we can use different methods such as drawing their nets, creating physical models using paper or clay, or using computer software to create 3D representations. Visualization helps us understand the characteristics and properties of solid shapes.
3. What is the importance of visualizing solid shapes?
Ans. Visualizing solid shapes is important as it helps us understand the spatial relationships, dimensions, and properties of objects in our surroundings. It is a crucial skill in fields such as architecture, engineering, and design, where accurate visualization of shapes is essential for creating and understanding structures and objects.
4. Can solid shapes be classified based on their properties?
Ans. Yes, solid shapes can be classified based on their properties. Some common classification criteria include the number of faces, edges, and vertices they have, as well as their symmetry, regularity, and the presence of parallel or perpendicular surfaces. This classification helps us categorize and study different types of solid shapes.
5. How can we find the surface area and volume of solid shapes?
Ans. The surface area of a solid shape is the sum of the areas of all its faces, while the volume is the amount of space enclosed by the shape. The formulas for finding the surface area and volume vary depending on the type of solid shape. For example, the surface area of a cube is given by 6 times the square of its edge length, while the volume of a cylinder is calculated by multiplying the base area with the height.
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