## RD Sharma Solutions for Class 8 Chapter 19 Visualising Shapes Free Online

EXERCISE 19.1 PAGE NO: 19.9

**1. What is the least number of planes that can enclose a solid? What is the name of the solid?**

**Solution:**

The least number of planes that are required to enclose a solid is 4.

The name of solid is tetrahedron.

**2. Can a polyhedron have for its faces?**

(i) 3 triangles?

(ii) 4 triangles?

(iii) a square and four triangles?

(i) 3 triangles?

(ii) 4 triangles?

(iii) a square and four triangles?

**Solution:**

(i) 3 triangles?

No, because a polyhedron is a solid shape bounded by polygons.

(ii) 4 triangles?

Yes, because a tetrahedron as 4 triangles as its faces.

(iii) a square and four triangles?

Yes, because a square pyramid has a square and four triangles as its faces.

**3. Is it possible to have a polyhedron with any given number of faces?**

**Solution:**

Yes, if number of faces is four or more.

**4. Is a square prism same as a cube?**

**Solution:**

Yes. We know that a square is a three dimensional shape with six rectangular shaped sides, out of which two are squares. Cubes are of rectangular prism length, width and height of same measurement.

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

**Solution:**

No.

Let us use Euler’s formula

V + F = E + 2

15 + 10 = 20 + 2

25 ≠ 22

Since the given polyhedron is not following Euler’s formula, therefore it is not possible to have 10 faces, 20 edges and 15 vertices.

**6. Verify Euler’s formula for each of the following polyhedrons:**

**Solution:**

**(i)**Vertices = 10

Faces = 7

Edges = 15

By using Euler’s formula

V + F = E + 2

10 + 7 = 15 + 2

17 = 17

Hence verified.

**(ii)**Vertices = 9

Faces = 9

Edges = 16

By using Euler’s formula

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Hence verified.

**(iii)**Vertices = 14

Faces = 8

Edges = 20

By using Euler’s formula

V + F = E + 2

14 + 8 = 20 + 2

22 = 22

Hence verified.

**(iv)**Vertices = 6

Faces = 8

Edges = 12

By using Euler’s formula

V + F = E + 2

6 + 8 = 12 + 2

14 = 14

Hence verified.

**(v)**Vertices = 9

Faces = 9

Edges = 16

By using Euler’s formula

V + F = E + 2

9 + 9 = 16 + 2

18 = 18

Hence verified.

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

Faces | ? | 5 | 20 |

Vertices | 6 | ? | 12 |

Edges | 12 | 9 | ? |

**Solution:**

**(i)**

By using Euler’s formula

V + F = E + 2

6 + F = 12 + 2

F = 14 – 6

F = 8

∴ Number of faces is 8

**(ii)**

By using Euler’s formula

V + F = E + 2

V + 5 = 9 + 2

V = 11 – 5

V = 6

∴ Number of vertices is 6

**(iii)**

By using Euler’s formula

V + F = E + 2

12 + 20 = E + 2

E = 32 – 2

E = 30

∴ Number of edges is 30

EXERCISE 19.2 PAGE NO: 19.12

**1. Which among of the following are nets for a cube?**

**Solution:**

Figure (iv), (v), (vi) are the nets for a cube.

**2. Name the polyhedron that can be made by folding each net:**

**Solution:**

(i) From figure (i), a Square pyramid can be made by folding each net.

(ii) From figure (ii), a Triangular prism can be made by folding each net.

(iii) From figure (iii), a Triangular prism can be made by folding each net.

(iv) From figure (iv), a Hexagonal prism can be made by folding each net.

(iv) From figure (v), a Hexagonal pyramid can be made by folding each net.

(v) From figure (vi), a Cube can be made by folding each net.

**3. Dice are cubes where the numbers on the opposite faces must total 7. Which of the following are dice?**

**Solution:**

Figure (i), is a dice. Since the sum of numbers on opposite faces is 7 (3 + 4 = 7 and 6 + 1 = 7).

**4. Draw nets for each of the following polyhedrons:**

**Solution:**

**(i)**The net pattern for cube is

**(ii)**The pattern for triangular prism is

**(iii)**The net pattern for hexagonal prism is

**(iv)**The net pattern for pentagonal pyramid is

**5. Match the following figures:**

**Solution:**

(a)-(iv) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 4×4 = 16

(b)-(i) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 8×3 = 24

(c)-(ii) Because multiplication of numbers on adjacent faces are equal, where 6×4 = 24 and 6×3 = 18

(d)-(iii) Because multiplication of numbers on adjacent faces are equal, where 3×3 = 9 and 3×9 = 27