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Class 8 SELINA Solutions Maths Chapter 19: Representing 3-D in 2-D

Representing 3-D in 2-D Exercise Ex. 19

Solution 1(i)

Correct option: (b) 4

A tetrahedron has 4 faces.

Solution 1(ii)

Correct option: (b) 8

An octahedron has 8 faces.

Solution 1(iii)

Correct option: (d) 5 vertices and 5 faces

A rectangular pyramid has 5 vertices and 5 faces.

Solution 1(iv)

Correct option: (c) F + V – E = 2

Euler’s formula  is F + V – E = 2

Solution 1(v)

Correct option: (c) 30

Number of faces, F = 12

Number of vertices, V = 20

Then, by Euler’s formula,

F + V – E = 2

⇒ 12 + 20 – E = 2

⇒ E = 30

Solution 2

Number of faces, F = 7

Number of vertices, V = 10

Number of edges = E

Then, by Euler’s formula,

F + V – E = 2

⇒ 7 + 10 – E = 2

⇒ E = 15

Solution 3

(i) Pentagonal pyramid

      Number of faces = 6

      Number of vertices = 6

      Number of edges = 10

(ii) Hexagonal prism

      Number of faces = 8

      Number of vertices = 12

      Number of edges = 18

Solution 4(i)

For a given figure,

Number of faces, F = 5

Number of vertices, V = 5

Number of edges, E = 8

Now, F + V – E = 5 + 5 – 8 = 2

Hence, verified.

Solution 4(ii)

For a given figure,

Number of faces, F = 9

Number of vertices, V = 9

Number of edges, E = 16

Now, F + V – E = 9 + 9 – 16 = 2

Hence, verified.

Solution 4(iii)

For a given figure,

Number of faces, F = 7

Number of vertices, V = 10

Number of edges, E = 15

Now, F + V – E = 7 + 10 – 15 = 2

Hence, verified.

Solution 5

Number of faces, F = 8

Number of vertices, V = 16

Number of edges, E = 26

Now, F + V – E = 8 + 16 – 26 = –2 ≠ 2

Therefore, a polyhedron cannot have 8 faces, 26 edges and 16 vertices.

Solution 6

(i) No, a polyhedron is a three-dimensional figure with at least 4 polygonal faces.

(ii) Yes, a polyhedron can have 4 triangles only. A tetrahedron has 4 triangular faces.

(iii) Yes, a polyhedron can have a square and four triangles. A square pyramid has a square and four triangular faces.

Solution 7

Euler’s formula: F + V – E = 2

(i) F = x, V = 15, E = 20

x + 15 – 20 = 2

x – 5 = 2

x = 7

(ii) F = 6, V = y, E = 8

6 + y – 8 = 2

y – 2 = 2

y = 4

(iii) F = 14, V = 26, E = z

14 + 26 – z = 2

40 – z = 2

z = 38

Solution 8

The least number of planes that can enclose a solid is 4.

The name of the solid is tetrahedron.

Solution 9

Yes, a square prism is same as a cube.

Solution 10

Two nets of cuboid with dimensions 6 cm × 4 cm × 2 cm are as follows:

Solution 11

The sum of opposite faces of dice is 7.

Here,

Opposite face of a will be 5.

⇒ a + 5 = 7

⇒ a = 2

Opposite face of b will be 6.

⇒ b + 6 = 7

⇒ b = 1

Opposite face of c will be is 4.

⇒ c + 4 = 7

⇒ c = 3

Solution 12

 Nets for Hexagonal prism:

           

Nets for Pentagonal pyramid:

       

 

Representing 3-D in 2-D Exercise TEST YOURSELF

Solution 1(i)

Correct option: (a) 12

By Euler’s formula, F + V – E = 2

⇒ 20 + V – 30 = 2

⇒ V – 10 = 2

⇒ V = 12

Solution 1(ii)

Correct option: (b) triangular prism

It is a net of triangular prism.

Solution 1(iii)

Correct option: (c) 15

A pentagonal prism has 15 edges. Hence, Joseph will need 15 straws.

Solution 1(iv)

Correct option: (c) 14

A hexagonal pyramid has 12 edges,

By Euler’s formula, F + V – E = 2

Therefore, F + V = 2 + E = 2 + 12 = 14

Solution 1(v)

Correct option: (b) 4

A triangular pyramid has 4 faces.

Solution 2

A net is prepared for three-dimensional solids. Since rectangle is a two-dimensional figure, it does not have a net. It can be drawn as follows:

Solution 3

Number of faces, F = 15

Number of vertices, V = 20

Number of edges, E = 30

Now, F + V – E = 15 + 20 – 30 = 5 ≠ 2

Therefore, a polyhedron cannot have 15 faces, 30 edges and 20 vertices.

Solution 4

The net of a square pyramid is as follows:

Solution 5

A hexagonal pyramid has 12 edges.

Solution 6

The two-dimensional representation of a triangular prism is as follows:

Solution 7

Euler’s formula: F + V – E = 2

For a given polyhedron, F = 10, V = 8, E = ?

10 + 8 – E = 2

18 – E = 2

E = 16

Hence, the number of edges is 16.

Solution 8

(i) Triangular prism

(ii) Triangular prism

(iii) Hexagonal pyramid