Class 8 | NCERT Solution Maths Chapter 10 | Visualising Solid Shapes | Exercise 10.3

Exercise 10.3 Page No: 166

1. Can a polyhedron have for its faces:

(i) 3 Triangles?

(ii) 4 triangles?

(iii) A square and four triangles?

Solution:

(i) No, such polyhedrons are not possible. Such figures should have minimum 4 faces.

(ii) Yes, a triangular pyramid has 4 triangular faces.

(iii) Yes, as square pyramid has a square face and 4 triangular faces.

2. Is it possible to have a polyhedron with any given number of faces? (Hint: Think of a pyramid)

Solution:

It is possible, only if the number of faces is greater than or equal to 4.

3. Which are prisms among the following:

Solution:

(i) A nail: Not a polyhedron as it has a curved surface. This is not a prism.

(ii) Unsharpened Pencil: It is a prism.

(iii) A table Weight: It is not a prism.

(iv) A Box: It is a prism.

4. (i) How are prisms and cylinders alike?

(ii) How are pyramids and cones alike?

Solution:

(i) A cylinder can be looks like circular prism, a prism with circular base.

(ii) A cone can be a circular pyramid, a pyramid with circular base.

5. Is a square prism same as a cube? Explain.

Solution:

Yes, a square prism can also be a cube. A square prism has a square as its base. However, its height is not necessarily same as the side of the square. Thus, a square prism can also be a cuboid.

6. Verify Euler’s formula for these solids.

Solution:

(i) Number of faces, F = 7

Number of edges, E = 15

Number of vertices, V = 10

As per formula, F + V - E = 2

Substitute the values, we have

F + V - E = 7 + 10 - 15

= 2

Hence, verified.

(ii) Here, F = 9, V = 9 and E = 16

Using formula, F + V - E = 2

F + V - E = 9 + 9 - 16 = 2

Hence, Euler’s formula is verified.

7. Using Euler’s formula, find the unknown:

Solution:

Euler's formula: F + V - E = 2

Where, F = Faces, V = Vertices and E = Edges

(i) F + 6 – 12 = 2

F = 2 + 6

⇒ F = 8

(ii) 5 + V - 9 = 2

V - 4 = 2

⇒ V = 6

(iii) 20 + 12 - E = 2

32 – E = 2

⇒ E = 30

8. Can a polyhedron have 10 faces, 20 edges and 15 vertices?

Solution:

From the given data, we have

F = 10

E = 20

V = 15

Every polyhedron satisfies Euler’s formula, which is stated as, F + V – E = 2

For the given polygon,

F + V – E = 10 + 15 – 20 = 25 – 20 = 5, which is not equal to 2

Therefore, A polyhedron is not possible as Euler’s formula is not satisfied.