Representing 3-D in 2-D

ICSE Class 8 Mathematics · Selina Solutions Concise Mathematics Class 8 ICSE · 34 solved questions

Exercise

1(i)Exercise 19

Number of faces in a tetrahedron is :

(a)3
(b)4
(c)5
(d)6

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Number of faces in a tetrahedron is 4.

Hence, option 2 is the correct option.

1(ii)Exercise 19

In an octahedron; number of faces is :

(a)half of the number of edges
(b)8
(c)12
(d)more than the number of edges

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Number of faces in an octahedron is 8.

Hence, option 2 is the correct option.

1(iii)Exercise 19

A rectangular pyramid has :

(a)4 vertices and 4 faces
(b)6 vertices and 6 faces
(c)7 vertices and 4 faces
(d)5 vertices and 5 faces

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A rectangular pyramid has 5 vertices and 5 faces.

Hence, option 4 is the correct option.

1(iv)Exercise 19

If V stands for number of vertices, F stands for number of faces and E stands for number of edges, then :

(a)V + F + E = 2
(b)F + E - V - 2 = 0
(c)F + V - E = 2
(d)F - E + V + 2 = 0

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According to Euler's formula,

F + V - E = 2

Hence, option 3 is the correct option.

1(v)Exercise 19

If in a polyhedron, number of faces = 12 and number of vertices is 20, then the number of edges is :

(a)10
(b)20
(c)30
(d)40

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According to Euler's formula,

F + V - E = 2

Given: F = 12

V = 20

So,

⇒ 12 + 20 - E = 2

⇒ 32 - E = 2

⇒ E = 32 - 2

⇒ E = 30

The number of edges = 30.

Hence, option 3 is the correct option.

2Exercise 19

If a polyhedron, has 10 vertices and 7 faces, find the number of edges in it.

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Given:

Vertices, V = 10

Faces, F = 7

According to Euler's formula,

F + V - E = 2

So,

⇒ 7 + 10 - E = 2

⇒ 17 - E = 2

⇒ E = 17 - 2

⇒ E = 15

Hence, number of edges = 15.

3Exercise 19

State the number of faces, number of vertices and the number of edges of :

(i) a pentagonal pyramid

(ii) a hexagonal prism

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(i) A pentagonal pyramid

Number of faces = 6

Number of vertices = 6

Number of edges = 10

(ii) A hexagonal prism

Number of faces = 8

Number of vertices = 12

Number of edges = 18

4Exercise 19

Verify Euler's formula for the following three dimensional figures :

(i)

(ii)

(iii)

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(i)

Given:

F = 8, V = 6 , E = 12

According to Euler's formula,

F + V - E = 2

So, LHS:

⇒ 8 + 6 - 12

⇒ 14 - 12

⇒ 2

Thus, LHS = RHS

Hence, Euler's formula is verified.

(ii)

Given:

F = 9, V = 9 , E = 16

According to Euler's formula,

F + V - E = 2

So, LHS:

⇒ 9 + 9 - 16

⇒ 18 - 16

⇒ 2

Thus, LHS = RHS

Hence, Euler's formula is verified.

(iii)

Given:

F = 7, V = 10 , E = 15

According to Euler's formula,

F + V - E = 2

So, LHS:

⇒ 7 + 10 - 15

⇒ 17 - 15

⇒ 2

Thus, LHS = RHS

Hence, Euler's formula is verified.

5Exercise 19

Can a polyhedron have 8 faces, 26 edges and 16 vertices ?

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Clearly, F = 8

V = 16

E = 26

According to Euler's formula,

F + V - E = 2

F + V - E = 8 + 16 - 26

= 24 - 26

= - 2

⇒ F + V - E is not equal to 2.

Hence, a polyhedron can not have 8 faces, 26 edges and 16 vertices.

6Exercise 19

Can a polyhedron have :

(i) 3 triangles only ?

(ii) 4 triangles only ?

(iii) a square and four triangles?

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(i) No, a polyhedron cannot have only 3 triangles as its faces because a polyhedron must have at least four faces.

(ii) Yes, a polyhedron can have 4 triangles, which is known as a pyramid with a triangular base, because all the edges meet at the vertices.

(iii) Yes, a polyhedron can have a square and four triangles as its faces, forming a pyramid with a square base, because all eight edges meet at the vertices.

7Exercise 19

Using Euler's formula, find the values of x, y and z

S.No	Faces	Vertices	Edges
(i)	x	15	20
(ii)	6	y	8
(iii)	14	26	z

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(i) F + V - E = 2

⇒ x + 15 - 20 = 2

⇒ x - 5 = 2

⇒ x = 2 + 5

⇒ x = 7

Hence, x = 7.

(ii) F + V - E = 2

⇒ 6 + y - 8 = 2

⇒ y - 2 = 2

⇒ y = 2 + 2

⇒ y = 4

Hence, y = 4.

(iii) F + V - E = 2

So,

⇒ 14 + 26 - z = 2

⇒ 40 - z = 2

⇒ z = 40 - 2

⇒ z = 38

Hence, z = 38.

S.No	Faces	Vertices	Edges
(i)	7	15	20
(ii)	6	4	8
(iii)	14	26	38

8Exercise 19

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

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The least number of planes that can enclose a solid is four. The solid formed by these four planes is called a tetrahedron.

Hence, the shape is tetrahedron which has 4 planes.

9Exercise 19

Is a square prism same as a cube ?

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A cube is a three-dimensional solid shape with equal length, width and height.

A square prism has a square base, but its height may not be the same as its base dimensions.

If the height is equal to the base's dimensions, then the square prism is a cube.
If the height differs from the base's dimensions, then the square prism is a cuboid.

Hence, a square prism can either be a cube or not.

10Exercise 19

The dimensions of a cuboid are 6 cm x 4 cm x 2 cm. Draw two different nets of it.

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Two different nets of the cuboid are shown below:

Net 1

The dimensions of a cuboid are 6 cm x 4 cm x 2 cm. Draw two different nets of it. Representing 3-D in 2-D, Concise Mathematics Solutions ICSE Class 8.

Net 2

11Exercise 19

Dice are cubes where the sum of the numbers on the opposite faces is 7. Find the missing numbers a, b and c.

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It is given that the sum of the numbers on the opposite faces is 7.

So, c + 4 = 7

⇒ c = 7 - 4

⇒ c = 3

And, 5 + a = 7

⇒ a = 7 - 5

⇒ a = 2

And, 6 + b = 7

⇒ b = 7 - 6

⇒ b = 1

Hence, the values are a = 2, b = 1 and c = 3.

12(i)Exercise 19

Draw two nets for the following polyhedron :

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The two nets are shown below:

12(ii)Exercise 19

Draw two nets for the following polyhedron :

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The two nets are shown below:

1(i)Test Yourself

If in a polyhedron, number of faces = 20 and number edges = 30; the number of vertices is :

(a)12
(b)6
(c)8
(d)20

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Given:

F = 20, E = 30

According to Euler's formula,

F + V - E = 2

⇒ 20 + V - 30 = 2

⇒ V - 10 = 2

⇒ V = 2 + 10

⇒ V = 12

Hence, option 1 is the correct option.

1(ii)Test Yourself

is the net of :

(a)square pyramid
(b)triangular prism
(c)hexagonal pyramid
(d)none of these

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The given figure is the net of triangular prism.

Hence, option 2 is the correct option.

1(iii)Test Yourself

Joseph is making a pentagonal prism using identical straws. How many straws does he need ?

(a)20
(b)18
(c)15
(d)10

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The number of edges in a pentagonal prism is 15, so Joseph needs 15 identical straws.

Hence, option 3 is the correct option.

1(iv)Test Yourself

In case of a hexagonal pyramid, F = no. of faces and V = no. of vertices, then F + V is:

(a)10
(b)12
(c)14
(d)16

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For a hexagonal pyramid,

F = 7, V = 7

So, F + V = 7 + 7

= 14

Hence, option 3 is the correct option.

1(v)Test Yourself

The number of faces in a triangular pyramid is :

(a)3
(b)4
(c)5
(d)none of these

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Number of faces in a triangular pyramid is 4.

Hence, option 2 is the correct option.

1(vi)Test Yourself

Statement 1: The number of vertices in a pyramid is one more than the number of sides in a polygon.

Statement 2: A polyhedron may have 10 faces, 20 edges and 15 vertices.

Which of the following options is correct?

(a)Both the statements are true.
(b)Both the statements are false.
(c)Statement 1 is true, and statement 2 is false.
(d)Statement 1 is false, and statement 2 is true.

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The base of a pyramid is a polygon. Let no. of sides in polygon be n, so total number of vertices in a polygon will be n.

So, the total number of vertices in a pyramid will be n + 1.

According to statement 1 :

The number of vertices in a pyramid is one more than the number of sides in a polygon.

So, statement 1 is true.

Given,

A polyhedron may have 10 faces, 20 edges and 15 vertices.

If this is true, then it will satisfy Euler's formula.

Using Euler's formula :

F + V - E = 2.

Substituting the values in L.H.S., we get

⇒ 10 + 15 - 20

⇒ 25 - 20

⇒ 5

R.H.S. = 2

Since, L.H.S. ≠ R.H.S.

So, statement 2 is false.

∴ Statement 1 is true, and statement 2 is false.

Hence, option 3 is the correct option.

1(vii)Test Yourself

Assertion (A) : In a polyhedron, if there are 6 vertices, 12 edges, then the number of faces are 8.

Reason (R) : In a pentagonal pyramid, there are 6 faces, 6 vertices and 10 edges.

(a)Both A and R are correct, and R is the correct explanation for A.
(b)Both A and R are correct, and R is not the correct explanation for A.
(c)A is true, but R is false.
(d)A is false, but R is true.

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In assertion (A),

Given,

E = 12 and V = 6.

Using Euler's formula :

⇒ F + V - E = 2.

Substituting the values, we get :

⇒ F + 6 - 12 = 2

⇒ F - 6 = 2

⇒ F = 2 + 6

⇒ F = 8

So, assertion (A) is true.

In reason (R),

Given,

F = 6, E = 10, and V = 6.

Using Euler's formula :

⇒ F + V - E = 2.

Substituting the values in L.H.S., we get

⇒ 6 + 6 - 10

⇒ 12 - 10

⇒ 2

R.H.S. = 2

Since, L.H.S. = R.H.S.

So, reason (R) is true.

∴ Both A and R are correct, and R is not the correct explanation for A.

Hence, option 2 is the correct option.

1(viii)Test Yourself

Assertion (A) : If a polyhedron has 7 vertices and 10 faces, the number of edges is 19.

Reason (R) : The relationship between faces (F), edges (E) and vertices (V) of a polyhedron is F + V - E = 2.

(a)Both A and R are correct, and R is the correct explanation for A.
(b)Both A and R are correct, and R is not the correct explanation for A.
(c)A is true, but R is false.
(d)A is false, but R is true.

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By Euler's formula, the relationship between faces (F), edges (E) and vertices (V) of a polyhedron is : F + V - E = 2.

So, reason (R) is true.

Given,

V = 7 and F = 10

Substituting the values, in Euler's formula we get :

⇒ 10 + 7 - E = 2

⇒ 17 - E = 2

⇒ E = 17 - 2

⇒ E = 15

So, assertion (A) is false.

∴ A is false, but R is true.

Hence, option 4 is the correct option.

1(ix)Test Yourself

Assertion (A) : The number of edges in a triangular prism = 9.

Reason (R) : In a triangular prism, the number of vertices = 2 x number of sides = 6;

The number of faces = 2 + number of sides = 5.

(a)Both A and R are correct, and R is the correct explanation for A.
(b)Both A and R are correct, and R is not the correct explanation for A.
(c)A is true, but R is false.
(d)A is false, but R is true.

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A triangular prism has 3 edges on each triangular base and 3 edges connecting the bases.

Total edges = 3 + 3 + 3 = 9

So, assertion (A) is true.

As we know that

A triangular prism has 5 faces, 9 sides, and 6 vertices.

The number of vertices = 2 x number of sides = 2 x 9 = 18.

The number of faces = 2 + number of sides = 2 + 9 = 11.

So, reason (R) is false.

∴ A is true, but R is false.

Hence, option 3 is the correct option.

1(x)Test Yourself

Assertion (A) : The number of edges in a rectangular pyramid = 8.

Reason (R) : In a triangular prism, the number of vertices is one more than number of sides and the number of faces is one less than number of sides.

(a)Both A and R are correct, and R is the correct explanation for A.
(b)Both A and R are false.
(c)A is true, but R is false.
(d)A is false, but R is true.

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A rectangular pyramid has 5 faces, 8 edges, and 5 vertices.

So, assertion (A) is true.

A triangular prism has 5 faces, 9 sides, and 6 vertices.

Difference between sides and vertices = 9 - 6 = 3

Difference between sides and faces = 9 - 5 = 4

So, reason (R) is false.

∴ A is true, but R is false.

Hence, option 3 is the correct option.

2Test Yourself

Draw the net of a rectangular box.

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Net of a rectangular box is shown below:

Draw the net of a rectangular box. Representing 3-D in 2-D, Concise Mathematics Solutions ICSE Class 8.

3Test Yourself

Can a polyhedron have 15 faces, 30 edges and 20 vertices ?

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F = 15, V = 20, E = 30

According to Euler's formula,

F + V - E = 2

F + V - E = 15 + 20 - 30

= 35 - 30

= 5

⇒ F + V - E is not equal to 2.

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

4Test Yourself

Draw the net of a square pyramid.

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Net of a square pyramid is shown below:

Draw the net of a square pyramid. Representing 3-D in 2-D, Concise Mathematics Solutions ICSE Class 8.

5Test Yourself

Write the number of edges in a hexagonal pyramid.

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The number of edges in a hexagonal pyramid is 12.

6Test Yourself

Draw the two dimensional representation of a triangular prism.

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Two dimensional representation of a triangular prism is shown below:

Draw the two dimensional representation of a triangular prism. Representing 3-D in 2-D, Concise Mathematics Solutions ICSE Class 8.

7Test Yourself

If a polyhedron has 10 faces and 8 vertices, find the number of edges in it.

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Given:

F = 10, V = 8

According to Euler's formula,

F + V - E = 2

⇒ 10 + 8 - E = 2

⇒ 18 - E = 2

⇒ E = 18 - 2

⇒ E = 16

Hence, the number of edges = 16.

8Test Yourself

Name the polyhedron that can be made by folding each of the following nets :

(i)

(ii)

(iii)

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(i) It has 3 rectangles and 2 triangles.

Hence, it is triangular prism.

(ii) It has 3 rectangles and 2 triangles.

Hence, it is triangular prism.

(iii) It has 1 hexagonal base and 6 triangles.

Hence, it is hexagonal pyramid.

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