Visualising Solid Shapes

ICSE Class 8 Mathematics · ML Aggarwal Solutions Understanding ICSE Mathematics Class 8 · 52 solved questions

Exercise

Marks

1(i)Exercise 18.1

Draw an oblique sketch of a cube of side 3 units.

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Steps of construction :

Take a squared paper of side 1 unit.
Draw a square of side 3 units to get the front face.

Draw an oblique sketch of a cube of side 3 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw the opposite face overlapping the first square. This square is 1 unit to the right and 1 unit above the previous square.

Join the corresponding corners of the two squares by oblique lines to get the required cube.

1(ii)Exercise 18.1

Draw an isometric sketch of a cube of side 3 units.

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Steps of construction :

Take an isometric dot paper of side 1 unit.
Draw two base edges, each of length 3 units, along the two isometric directions from a common vertex.

Draw an isometric sketch of a cube of side 3 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw three vertical lines each of height 3 units, one on the middle vertex and the other two on the end points of these line segments.

For the top face, draw a rhombus of side 3 units whose adjacent sides are formed by joining the top free ends of the three vertical line segments to get the required cube.

2(i)Exercise 18.1

Draw an oblique sketch of a cuboid of length 5 units, breadth 3 units and height 4 units.

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Steps of construction :

Take a squared paper of side 1 unit.
Draw a rectangle of dimensions 5 units × 4 units to get the front face of the cuboid.

Draw an oblique sketch of a cuboid of length 5 units, breadth 3 units and height 4 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw the opposite face of the same size overlapping the first rectangle, such that it is 1 unit to the right and 1 unit above the first rectangle.

Join the corresponding corners of the two rectangles by oblique lines to get the required cuboid.

2(ii)Exercise 18.1

Draw an isometric sketch of a cuboid of length 5 units, breadth 3 units and height 4 units.

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Steps of construction :

Take an isometric dot paper of side 1 unit.
From a common vertex, draw two base edges of lengths 5 units and 3 units along the two isometric directions.

Draw an isometric sketch of a cuboid of length 5 units, breadth 3 units and height 4 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw three vertical lines each of height 4 units, one on the middle vertex and the other two on the end points of these line segments.

For the top face, draw a parallelogram of length 5 units and breadth 3 units whose adjacent sides are formed by joining the top free ends of the three vertical line segments to get the required cuboid.

3Exercise 18.1

Draw an oblique sketch of a square pyramid of side 3 units and height 4 units.

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Steps of construction :

Take a squared paper of side 1 unit.
Draw a square ABCD of side 3 units to get the base of the pyramid.

Draw an oblique sketch of a square pyramid of side 3 units and height 4 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw the diagonals AC and BD of this square and let them meet at the point N.

Through N, draw a vertical line segment ON of length 4 units, where O is the vertex of the pyramid.

Join OA, OB, OC and OD to get the required square pyramid.

4Exercise 18.1

Draw an isometric sketch of a hexagonal prism of height 5 units and base a regular hexagon of side 2 units.

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Steps of construction :

Take an isometric dot paper of side 1 unit.
Draw three consecutive sides of the base hexagon, each of length 2 units.

Draw an isometric sketch of a hexagonal prism of height 5 units and base a regular hexagon of side 2 units. Visualising Solid Shapes, ML Aggarwal Understanding Mathematics Solutions ICSE Class 8

Draw four vertical lines each of height 5 units, one on each of the two middle vertices and the other two on the end points of these line segments.

Using the four upper endpoints as four vertices, complete a regular hexagon of side 2 units to obtain the upper face.

1Exercise 18.2

Can a polyhedron have for its faces

(i) 3 triangles?

(ii) 4 triangles?

(iii) a square and four triangles?

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(i) A polyhedron must have at least 4 faces, so it cannot be enclosed by only 3 triangles.

Hence, a polyhedron cannot have 3 triangles for its faces.

(ii) A polyhedron can have 4 triangles as its faces. One of these triangles forms its base and the remaining three form its lateral faces meeting at a common vertex. Such a polyhedron is a triangular pyramid, which is also known as a tetrahedron.

Hence, a polyhedron can have 4 triangles for its faces.

(iii) A polyhedron can have a square and four triangles as its faces. The square forms its base and the four triangles form its lateral faces meeting at a common vertex. Such a polyhedron is a square pyramid.

Hence, a polyhedron can have a square and four triangles for its faces.

2Exercise 18.2

Which are prisms among the following?

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A prism is a polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape.

(i) A nail tapers to a pointed end, so its base and top are not congruent polygons. So, a nail is not a prism.

(ii) An unsharpened pencil has congruent hexagons as its base and top and rectangles as its lateral faces. So, an unsharpened pencil is a prism. It is a hexagonal prism.

(iii) A table weight has a square as its base and its lateral faces are triangles meeting at a common vertex. So, a table weight is not a prism. It is a square pyramid.

(iv) A box has congruent rectangles as its base and top and rectangles as its lateral faces. So, a box is a prism. It is a rectangular prism.

Hence, only (ii) and (iv) are prisms.

3(i)Exercise 18.2

Verify Euler's formula for this solid:

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The given solid has a pentagonal top, a pentagonal base and 5 lateral faces.

From the figure,

F = 7, V = 10, E = 15

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 7 + 10 = 17

R.H.S. = E + 2 = 15 + 2 = 17

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

Hence, Euler's formula is verified.

3(ii)Exercise 18.2

Verify Euler's formula for this solid:

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The given solid is a square prism surmounted by a square pyramid.

From the figure,

F = 9, V = 9, E = 16

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 9 + 9 = 18

R.H.S. = E + 2 = 16 + 2 = 18

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

Hence, Euler's formula is verified.

3(iii)Exercise 18.2

Verify Euler's formula for this solid:

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The given solid is a hexagonal pyramid.

From the figure,

F = 7, V = 7, E = 12

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 7 + 7 = 14

R.H.S. = E + 2 = 12 + 2 = 14

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

Hence, Euler's formula is verified.

3(iv)Exercise 18.2

Verify Euler's formula for this solid:

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The given solid is an octagonal pyramid.

From the figure,

F = 9, V = 9, E = 16

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 9 + 9 = 18

R.H.S. = E + 2 = 16 + 2 = 18

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

Hence, Euler's formula is verified.

4Exercise 18.2

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

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Given:

F = 15, E = 30, V = 20

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 15 + 20 = 35

R.H.S. = E + 2 = 30 + 2 = 32

Since, L.H.S. ≠ R.H.S.

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

5Exercise 18.2

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

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Given:

F = 8, V = 8

According to Euler's formula,

F + V = E + 2

⇒ 8 + 8 = E + 2

⇒ 16 = E + 2

⇒ E = 16 - 2

⇒ E = 14

Hence, the number of edges = 14.

6Exercise 18.2

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

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Given:

F = 7, V = 10

According to Euler's formula,

F + V = E + 2

⇒ 7 + 10 = E + 2

⇒ 17 = E + 2

⇒ E = 17 - 2

⇒ E = 15

Hence, the number of edges = 15.

7Exercise 18.2

Write the number of faces, vertices and edges in

(i) an octagonal prism

(ii) decagonal pyramid.

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(i) In an octagonal prism, the number of sides of the polygonal base is 8.

Number of faces in a prism is 2 more than the number of sides of the polygonal base.

Number of faces = 8 + 2 = 10

Number of vertices in a prism is twice the number of sides of the polygonal base.

Number of vertices = 8 × 2 = 16

Number of edges in a prism is thrice the number of sides of the polygonal base.

Number of edges = 8 × 3 = 24

Hence, in an octagonal prism, F = 10, V = 16 and E = 24.

(ii) In a decagonal pyramid, the number of sides of the polygonal base is 10.

Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.

Number of faces = 10 + 1 = 11

Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.

Number of vertices = 10 + 1 = 11

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 10 × 2 = 20

Hence, in a decagonal pyramid, F = 11, V = 11 and E = 20.

8Exercise 18.2

Using Euler's formula, complete the following table:

	Faces	Vertices	Edges
(i)	6	—	12
(ii)	—	5	8
(iii)	14	24	—
(iv)	—	16	30
(v)	16	—	42
(vi)	19	19	—

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According to Euler's formula,

F + V = E + 2

(i) Given: F = 6, E = 12

⇒ 6 + V = 12 + 2

⇒ 6 + V = 14

⇒ V = 14 - 6

⇒ V = 8

Hence, V = 8.

(ii) Given: V = 5, E = 8

⇒ F + 5 = 8 + 2

⇒ F + 5 = 10

⇒ F = 10 - 5

⇒ F = 5

Hence, F = 5.

(iii) Given: F = 14, V = 24

⇒ 14 + 24 = E + 2

⇒ 38 = E + 2

⇒ E = 38 - 2

⇒ E = 36

Hence, E = 36.

(iv) Given: V = 16, E = 30

⇒ F + 16 = 30 + 2

⇒ F + 16 = 32

⇒ F = 32 - 16

⇒ F = 16

Hence, F = 16.

(v) Given: F = 16, E = 42

⇒ 16 + V = 42 + 2

⇒ 16 + V = 44

⇒ V = 44 - 16

⇒ V = 28

Hence, V = 28.

(vi) Given: F = 19, V = 19

⇒ 19 + 19 = E + 2

⇒ 38 = E + 2

⇒ E = 38 - 2

⇒ E = 36

Hence, E = 36.

The completed table is shown below:

	Faces	Vertices	Edges
(i)	6	8	12
(ii)	5	5	8
(iii)	14	24	36
(iv)	16	16	30
(v)	16	28	42
(vi)	19	19	36

1Objective Type Questions - Mental Maths

Fill in the blanks:

(i) A solid made up of polygonal regions is called .....

(ii) Polyhedrons which are not convex are called .....

(iii) A polyhedron is said to be ..... if all of its faces are regular polygons and the same number of faces meet at each vertex.

(iv) A polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape is called a .....

(v) A prism whose base and top are congruent hexagon is called a .....

(vi) A polyhedron whose all lateral faces are triangles is called a .....

(vii) A triangular pyramid is also known as .....

(viii) A rectangular prism is also known as .....

(ix) The polygonal regions forming a polyhedron are called .....

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(i) A solid made up of polygonal regions is called polyhedron.

(ii) Polyhedrons which are not convex are called concave polyhedrons.

(iii) A polyhedron is said to be regular polyhedron if all of its faces are regular polygons and the same number of faces meet at each vertex.

(iv) A polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape is called a prism.

(v) A prism whose base and top are congruent hexagon is called a hexagonal prism.

(vi) A polyhedron whose all lateral faces are triangles is called a pyramid.

(vii) A triangular pyramid is also known as tetrahedron.

(viii) A rectangular prism is also known as cuboid.

(ix) The polygonal regions forming a polyhedron are called faces.

2Objective Type Questions - Mental Maths

State which of the following statements are true (T) or false (F):

(i) A cylinder is a polyhedron.

(ii) All the prisms and pyramids are polyhedrons.

(iii) A tetrahedron is the only pyramid which can be a regular polyhedron.

(iv) The line segments where the faces of a polyhedron meet are called edges.

(v) F + E = V + 2 is called Euler's formula.

(vi) In any prism number of faces is 2 more than number of sides of polygonal base.

(vii) In any pyramid number of edges is twice the number of sides of polygonal base.

(viii) An octagonal prism has 18 vertices.

(ix) All pyramids are prisms.

(x) Lateral faces of a pyramid are triangles.

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(i) False

Reason:

A polyhedron is a solid made up of polygonal regions. A cylinder has a curved surface and two circular ends, so it is not made up of polygonal regions.

(ii) True

Reason:

The base, the top and the lateral faces of a prism, and the base and the lateral faces of a pyramid, are all polygonal regions. So, every prism and every pyramid is a polyhedron.

(iii) True

Reason:

In a tetrahedron all the four faces are congruent equilateral triangles and 3 faces meet at each of its 4 vertices. In every other pyramid, the base is not congruent to the lateral faces and the number of faces meeting at the apex is not the same as the number of faces meeting at a vertex of the base.

(iv) True

Reason:

The line segments where the faces of a polyhedron meet are called edges and the points where the edges of a polyhedron meet are called vertices.

(v) False

Reason:

Euler's formula states that in any polyhedron F + V = E + 2.

(vi) True

Reason:

A prism has one lateral face along each side of its polygonal base together with 2 more faces, which are its base and its top.

(vii) True

Reason:

A pyramid has one edge along each side of its polygonal base and one more edge joining each vertex of the base to the apex.

(viii) False

Reason:

The number of vertices in a prism is twice the number of sides of the polygonal base. So, an octagonal prism has 8 × 2 = 16 vertices.

(ix) False

Reason:

In a prism the base and the top are congruent polygons and the lateral faces are parallelograms in shape, whereas a pyramid has only one polygonal base and its lateral faces are triangles meeting at a common vertex.

(x) True

Reason:

A pyramid is a polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex.

31 Mark · Objective Type Questions - Multiple Choice…

Which of the following is not a 2-dimensional shape?

(a)Triangle
(b)Circle
(c)Sphere
(d)Rectangle

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A triangle, a circle and a rectangle are plane figures, so each of them is a 2-dimensional shape.

A sphere is a solid figure, so it is a 3-dimensional shape.

Hence, option 3 is the correct option.

41 Mark · Objective Type Questions - Multiple Choice…

Which of the following is a 3-dimensional shape?

(a)Parallelogram
(b)Cylinder
(c)Square
(d)none of these

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A parallelogram and a square are plane figures, so each of them is a 2-dimensional shape.

A cylinder is a solid figure, so it is a 3-dimensional shape.

Hence, option 2 is the correct option.

51 Mark · Objective Type Questions - Multiple Choice…

How many parallel and congruent faces does a cylinder have?

(a)4
(b)3
(c)2
(d)none

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A cylinder has two circular ends, which are parallel and congruent to each other.

So, the number of parallel and congruent faces of a cylinder = 2

Hence, option 3 is the correct option.

61 Mark · Objective Type Questions - Multiple Choice…

How many pairs of congruent parallel faces does a rectangular prism have?

(a)8
(b)6
(c)4
(d)3

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A rectangular prism is also known as a cuboid. It has 6 rectangular faces and its opposite faces are parallel and congruent to each other.

So, the number of pairs of congruent parallel faces = 6 ÷ 2 = 3

Hence, option 4 is the correct option.

71 Mark · Objective Type Questions - Multiple Choice…

How many congruent isosceles triangle does a square pyramid have?

(a)2
(b)4
(c)6
(d)8

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A square pyramid has a square as its base and 4 lateral faces, which are congruent isosceles triangles meeting at a common vertex.

So, the number of congruent isosceles triangles = 4

Hence, option 2 is the correct option.

81 Mark · Objective Type Questions - Multiple Choice…

Which amongst the following is not a polyhedron?

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A polyhedron is a solid made up of polygonal regions.

The solid given in option 3 is a cone. It has a circular base and a curved surface, so it is not made up of polygonal regions.

So, the solid given in option 3 is not a polyhedron.

Hence, option 3 is the correct option.

91 Mark · Objective Type Questions - Multiple Choice…

Which of the following is not a prism?

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A prism is a polyhedron whose base and top are congruent polygons and whose lateral faces are parallelograms in shape.

The solid given in option 2 tapers downwards, so its base and top are not congruent polygons.

So, the solid given in option 2 is not a prism.

Hence, option 2 is the correct option.

101 Mark · Objective Type Questions - Multiple Choice…

The number of triangular faces of a triangular prism are

(a)2
(b)3
(c)4
(d)5

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A triangular prism has 5 faces in all. Its base and top are congruent triangles and its 3 lateral faces are rectangles.

So, the number of triangular faces of a triangular prism = 2

Hence, option 1 is the correct option.

111 Mark · Objective Type Questions - Multiple Choice…

The number of edges in a pentagonal pyramid are

(a)5
(b)10
(c)15
(d)20

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In a pentagonal pyramid, the number of sides of the polygonal base is 5.

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 5 × 2 = 10

Hence, option 2 is the correct option.

121 Mark · Objective Type Questions - Multiple Choice…

The number of rectangular faces in a hexagonal prism are

(a)6
(b)8
(c)10
(d)12

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In a hexagonal prism, the number of sides of the polygonal base is 6.

Its base and top are congruent hexagons and it has one rectangular lateral face along each side of the base.

So, the number of rectangular faces = 6

Hence, option 1 is the correct option.

131 Mark · Objective Type Questions - Multiple Choice…

In a polyhedron E = 15, V = 10, then F is

(a)3
(b)5
(c)7
(d)9

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Given:

E = 15, V = 10

According to Euler's formula,

F + V = E + 2

⇒ F + 10 = 15 + 2

⇒ F + 10 = 17

⇒ F = 17 - 10

⇒ F = 7

Hence, option 3 is the correct option.

141 Mark · Objective Type Questions - Multiple Choice…

In a polyhedron F = 5, E = 8, then V is

(a)3
(b)5
(c)7
(d)9

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Given:

F = 5, E = 8

According to Euler's formula,

F + V = E + 2

⇒ 5 + V = 8 + 2

⇒ 5 + V = 10

⇒ V = 10 - 5

⇒ V = 5

Hence, option 2 is the correct option.

151 Mark · Objective Type Questions - Multiple Choice…

In a polyhedron F = 17, V = 30, then E is

(a)30
(b)45
(c)60
(d)none of these

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Given:

F = 17, V = 30

According to Euler's formula,

F + V = E + 2

⇒ 17 + 30 = E + 2

⇒ 47 = E + 2

⇒ E = 47 - 2

⇒ E = 45

Hence, option 2 is the correct option.

161 Mark · Objective Type Questions - Multiple Choice…

A polyhedron have 4 faces, 4 vertices and 6 edges. Name the polyhedron.

(a)A rectangular prism
(b)A triangular prism
(c)A rectangular pyramid
(d)A triangular pyramid

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A triangular pyramid has a triangle as its base and 3 triangles as its lateral faces, so F = 4.

Its base has 3 vertices and it has one more vertex at the apex, so V = 4.

It has 3 edges along its base and 3 more edges joining the vertices of the base to the apex, so E = 6.

So, the polyhedron having 4 faces, 4 vertices and 6 edges is a triangular pyramid.

Hence, option 4 is the correct option.

17Objective Type Questions - Statement I-II Type…

Statement I: A sphere is a regular polyhedron.

Statement II: A regular polyhedron has regular polygons as all its faces and same number of faces meet at each vertex.

(a)Statement I is true but Statement II is false.
(b)Statement I is false but Statement II is true.
(c)Both Statement I and Statement II are true.
(d)Both Statement I and Statement II are false.

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A sphere has a curved surface, so it is not made up of polygonal regions. So, a sphere is not a polyhedron and therefore it cannot be a regular polyhedron.

So, Statement I is false.

A polyhedron is said to be a regular polyhedron if all of its faces are regular polygons and the same number of faces meet at each vertex.

So, Statement II is true.

Hence, option 2 is the correct option.

18Objective Type Questions - Statement I-II Type…

Statement I: If a polyhedron has 5 faces and 15 edges, then it has 18 vertices.

Statement II: Euler's formula states that in any polyhedron F + E = V + 2

(a)Statement I is true but Statement II is false.
(b)Statement I is false but Statement II is true.
(c)Both Statement I and Statement II are true.
(d)Both Statement I and Statement II are false.

Show SolutionHide Solution↓

Given:

F = 5, E = 15

According to Euler's formula,

F + V = E + 2

⇒ 5 + V = 15 + 2

⇒ 5 + V = 17

⇒ V = 17 - 5

⇒ V = 12

So, the polyhedron has 12 vertices and not 18 vertices.

So, Statement I is false.

Euler's formula states that in any polyhedron F + V = E + 2 and not F + E = V + 2.

So, Statement II is false.

Hence, option 4 is the correct option.

19Objective Type Questions - Statement I-II Type…

Statement I: A heptagonal pyramid has 14 edges.

Statement II: A heptagonal prism has 21 edges.

(a)Statement I is true but Statement II is false.
(b)Statement I is false but Statement II is true.
(c)Both Statement I and Statement II are true.
(d)Both Statement I and Statement II are false.

Show SolutionHide Solution↓

In a heptagonal pyramid, the number of sides of the polygonal base is 7.

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 7 × 2 = 14

So, Statement I is true.

In a heptagonal prism, the number of sides of the polygonal base is 7.

Number of edges in a prism is thrice the number of sides of the polygonal base.

Number of edges = 7 × 3 = 21

So, Statement II is true.

Hence, option 3 is the correct option.

1(i)Check Your Progress

Write the number of faces, vertices and edges of a hexagonal pyramid.

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In a hexagonal pyramid, the number of sides of the polygonal base is 6.

Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.

Number of faces = 6 + 1 = 7

Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.

Number of vertices = 6 + 1 = 7

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 6 × 2 = 12

Hence, in a hexagonal pyramid, F = 7, V = 7 and E = 12.

1(ii)Check Your Progress

Write the number of faces, vertices and edges of an octagonal pyramid.

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In an octagonal pyramid, the number of sides of the polygonal base is 8.

Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.

Number of faces = 8 + 1 = 9

Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.

Number of vertices = 8 + 1 = 9

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 8 × 2 = 16

Hence, in an octagonal pyramid, F = 9, V = 9 and E = 16.

1(iii)Check Your Progress

Write the number of faces, vertices and edges of a decagonal pyramid.

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In a decagonal pyramid, the number of sides of the polygonal base is 10.

Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.

Number of faces = 10 + 1 = 11

Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.

Number of vertices = 10 + 1 = 11

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 10 × 2 = 20

Hence, in a decagonal pyramid, F = 11, V = 11 and E = 20.

1(iv)Check Your Progress

Write the number of faces, vertices and edges of a nonagonal pyramid.

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In a nonagonal pyramid, the number of sides of the polygonal base is 9.

Number of faces in a pyramid is 1 more than the number of sides of the polygonal base.

Number of faces = 9 + 1 = 10

Number of vertices in a pyramid is 1 more than the number of sides of the polygonal base.

Number of vertices = 9 + 1 = 10

Number of edges in a pyramid is twice the number of sides of the polygonal base.

Number of edges = 9 × 2 = 18

Hence, in a nonagonal pyramid, F = 10, V = 10 and E = 18.

1(v)Check Your Progress

Write the number of faces, vertices and edges of a heptagonal prism.

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In a heptagonal prism, the number of sides of the polygonal base is 7.

Number of faces in a prism is 2 more than the number of sides of the polygonal base.

Number of faces = 7 + 2 = 9

Number of vertices in a prism is twice the number of sides of the polygonal base.

Number of vertices = 7 × 2 = 14

Number of edges in a prism is thrice the number of sides of the polygonal base.

Number of edges = 7 × 3 = 21

Hence, in a heptagonal prism, F = 9, V = 14 and E = 21.

1(vi)Check Your Progress

Write the number of faces, vertices and edges of a decagonal prism.

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In a decagonal prism, the number of sides of the polygonal base is 10.

Number of faces in a prism is 2 more than the number of sides of the polygonal base.

Number of faces = 10 + 2 = 12

Number of vertices in a prism is twice the number of sides of the polygonal base.

Number of vertices = 10 × 2 = 20

Number of edges in a prism is thrice the number of sides of the polygonal base.

Number of edges = 10 × 3 = 30

Hence, in a decagonal prism, F = 12, V = 20 and E = 30.

2Check Your Progress

Give two examples of 3 dimensional shapes around you which are the combinations of 2 or more 3-dimensional shapes.

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Two such examples are given below:

A tomb — In a tomb, a cylinder is surmounted by a hemisphere.

An ice cream cone — In an ice cream cone, a cone is surmounted by a hemisphere.

Hence, a tomb and an ice cream cone are two shapes around us which are the combinations of 2 or more 3-dimensional shapes.

3Check Your Progress

Give two examples of solids which are not polyhedrons.

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A polyhedron is a solid made up of polygonal regions.

A sphere has a curved surface and a cylinder has a curved surface together with two circular ends. So, neither of them is made up of polygonal regions.

Hence, a sphere and a cylinder are two solids which are not polyhedrons.

4Check Your Progress

Why a pentagonal pyramid having all its edges congruent cannot be a regular polyhedron?

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A polyhedron is said to be a regular polyhedron if all of its faces are regular polygons and the same number of faces meet at each vertex.

In a pentagonal pyramid having all its edges congruent, the base is a regular pentagon and each of its 5 lateral faces is an equilateral triangle. So, all of its faces are regular polygons.

At the apex of this pyramid, 5 faces meet.

At each vertex of its base, 3 faces meet — two lateral faces and the base.

So, the same number of faces do not meet at each of its vertices.

Hence, a pentagonal pyramid having all its edges congruent cannot be a regular polyhedron, as all its vertices are not formed by the same number of faces.

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In a polyhedron, if F = 8 and V = 12 then find the number of edges.

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Given:

F = 8, V = 12

According to Euler's formula,

F + V = E + 2

⇒ 8 + 12 = E + 2

⇒ 20 = E + 2

⇒ E = 20 - 2

⇒ E = 18

Hence, the number of edges = 18.

6(i)Check Your Progress

Verify Euler's formula for the following figure:

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The given solid is a cuboid.

From the figure,

F = 6, V = 8, E = 12

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 6 + 8 = 14

R.H.S. = E + 2 = 12 + 2 = 14

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

Hence, Euler's formula is verified.

6(ii)Check Your Progress

Verify Euler's formula for the following figure:

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The given solid is a rectangular pyramid.

From the figure,

F = 5, V = 5, E = 8

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 5 + 5 = 10

R.H.S. = E + 2 = 8 + 2 = 10

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

Hence, Euler's formula is verified.

6(iii)Check Your Progress

Verify Euler's formula for the following figure:

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The given solid is a triangular pyramid.

From the figure,

F = 4, V = 4, E = 6

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 4 + 4 = 8

R.H.S. = E + 2 = 6 + 2 = 8

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

Hence, Euler's formula is verified.

6(iv)Check Your Progress

Verify Euler's formula for the following figure:

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The given solid is a pentagonal pyramid.

From the figure,

F = 6, V = 6, E = 10

According to Euler's formula,

F + V = E + 2

L.H.S. = F + V = 6 + 6 = 12

R.H.S. = E + 2 = 10 + 2 = 12

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

Hence, Euler's formula is verified.

7Check Your Progress

Which of the following nets can be folded to form a cone?

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The net of a cone consists of a circle, which forms its base, and a sector of a circle, which forms its curved surface.

In Figure 1 and Figure 2, the part which is to form the curved surface is cut off at its narrow end, so on folding it does not meet at a point. So, neither of these nets can be folded to form a cone.

In Figure 3, the part which is to form the curved surface tapers to a point and the circle attached to it forms the base. So, on folding, this net gives a cone.

Hence, the net given in Figure 3 can be folded to form a cone.

8Check Your Progress

Which of the following nets can be folded to form a cylinder?

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The net of a cylinder consists of a rectangle, which forms its curved surface, and two circles, which form its two ends.

On folding the net, the length of the rectangle becomes the circumference of each of the two circles. So, the two circles must be attached along the length of the rectangle.

In Figure 1 and Figure 3, the two circles are attached along the length of the rectangle. So, each of these nets can be folded to form a cylinder.

In Figure 2, the two circles are attached along the breadth of the rectangle. So, this net cannot be folded to form a cylinder.

Hence, the nets given in Figure 1 and Figure 3 can be folded to form a cylinder.

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