NCERT Textbook: Visualising Solid Shapes | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 198
Visualising Solid
Shapes
Chapter  13
13.1  INTRODUCTION: PLANE FIGURES AND SOLID SHAPES
In this chapter, you will classify figures you have seen in terms of what is known as
dimension.
In our day to day life, we see several objects like books, balls, ice-cream cones etc.,
around us which have different shapes. One thing common about most of these objects is
that they all have some length, breadth and height or depth.
That is, they all occupy space and have three dimensions.
Hence, they are called three dimensional shapes.
Do you remember some of the three dimensional shapes (i.e., solid shapes) we have
seen in earlier classes?
TRY THESE
Fig 13.1
(i) (a) Cuboid (iv) (d) Sphere
(ii) (b) Cylinder (v) (e) Pyramid
(iii) (c) Cube (vi) (f) Cone
Match the shape with the name:
2024-25
Page 2


MATHEMATICS 198
Visualising Solid
Shapes
Chapter  13
13.1  INTRODUCTION: PLANE FIGURES AND SOLID SHAPES
In this chapter, you will classify figures you have seen in terms of what is known as
dimension.
In our day to day life, we see several objects like books, balls, ice-cream cones etc.,
around us which have different shapes. One thing common about most of these objects is
that they all have some length, breadth and height or depth.
That is, they all occupy space and have three dimensions.
Hence, they are called three dimensional shapes.
Do you remember some of the three dimensional shapes (i.e., solid shapes) we have
seen in earlier classes?
TRY THESE
Fig 13.1
(i) (a) Cuboid (iv) (d) Sphere
(ii) (b) Cylinder (v) (e) Pyramid
(iii) (c) Cube (vi) (f) Cone
Match the shape with the name:
2024-25
VISUALISING SOLID SHAPES 199
Try to identify some objects shaped like each of these.
By a similar argument, we can say figures drawn on paper which have only length and
breadth are called two dimensional (i.e., plane) figures. We have also seen some two
dimensional figures in the earlier classes.
Match the 2 dimensional figures with the names (Fig 13.2):
(i) (a) Circle
(ii) (b) Rectangle
(iii) (c) Square
(iv) (d) Quadrilateral
(v) (e) Triangle
Fig 13.2
Note: We can write 2-D in short for 2-dimension and 3-D in short for
3-dimension.
13.2  FACES, EDGES AND VERTICES
Do you remember the Faces, Vertices and Edges of solid shapes, which you studied
earlier?  Here you see them for a cube:
         (i)            (ii)           (iii)
Fig 13.3
The 8 corners of the cube are its vertices. The 12 line segments that form the
skeleton of the cube are its edges. The 6 flat square surfaces that are the skin of the
cube are its faces.
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Page 3


MATHEMATICS 198
Visualising Solid
Shapes
Chapter  13
13.1  INTRODUCTION: PLANE FIGURES AND SOLID SHAPES
In this chapter, you will classify figures you have seen in terms of what is known as
dimension.
In our day to day life, we see several objects like books, balls, ice-cream cones etc.,
around us which have different shapes. One thing common about most of these objects is
that they all have some length, breadth and height or depth.
That is, they all occupy space and have three dimensions.
Hence, they are called three dimensional shapes.
Do you remember some of the three dimensional shapes (i.e., solid shapes) we have
seen in earlier classes?
TRY THESE
Fig 13.1
(i) (a) Cuboid (iv) (d) Sphere
(ii) (b) Cylinder (v) (e) Pyramid
(iii) (c) Cube (vi) (f) Cone
Match the shape with the name:
2024-25
VISUALISING SOLID SHAPES 199
Try to identify some objects shaped like each of these.
By a similar argument, we can say figures drawn on paper which have only length and
breadth are called two dimensional (i.e., plane) figures. We have also seen some two
dimensional figures in the earlier classes.
Match the 2 dimensional figures with the names (Fig 13.2):
(i) (a) Circle
(ii) (b) Rectangle
(iii) (c) Square
(iv) (d) Quadrilateral
(v) (e) Triangle
Fig 13.2
Note: We can write 2-D in short for 2-dimension and 3-D in short for
3-dimension.
13.2  FACES, EDGES AND VERTICES
Do you remember the Faces, Vertices and Edges of solid shapes, which you studied
earlier?  Here you see them for a cube:
         (i)            (ii)           (iii)
Fig 13.3
The 8 corners of the cube are its vertices. The 12 line segments that form the
skeleton of the cube are its edges. The 6 flat square surfaces that are the skin of the
cube are its faces.
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MATHEMATICS 200
Can you see that, the two dimensional figures can be identified as the faces of the
three dimensional shapes?  For example a cylinder  has two faces which are circles,
and a pyramid, shaped like this  has triangles as its faces.
We will now try to see how some of these 3-D shapes can be visualised on a 2-D
surface, that is, on paper.
In order to do this, we would like to get familiar with three dimensional objects closely .
Let us try forming these objects by making what are called nets.
13.3  NETS FOR BUILDING 3-D SHAPES
T ake a cardboard box.  Cut the edges to lay the box flat.  Y ou have now a net for that box.
A net is a sort of skeleton-outline in 2-D [Fig13.4 (i)], which, when folded [Fig13.4 (ii)],
results in a 3-D shape [Fig13.4 (iii)].
DO THIS
Vertex
Face
Edge
Face
Vertex
Edge
Faces (F) 6 4
Edges (E) 12
V ertices (V) 8 4
Complete the following table:
Table 13.1
(i) (ii) (iii)
Fig 13.4
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Page 4


MATHEMATICS 198
Visualising Solid
Shapes
Chapter  13
13.1  INTRODUCTION: PLANE FIGURES AND SOLID SHAPES
In this chapter, you will classify figures you have seen in terms of what is known as
dimension.
In our day to day life, we see several objects like books, balls, ice-cream cones etc.,
around us which have different shapes. One thing common about most of these objects is
that they all have some length, breadth and height or depth.
That is, they all occupy space and have three dimensions.
Hence, they are called three dimensional shapes.
Do you remember some of the three dimensional shapes (i.e., solid shapes) we have
seen in earlier classes?
TRY THESE
Fig 13.1
(i) (a) Cuboid (iv) (d) Sphere
(ii) (b) Cylinder (v) (e) Pyramid
(iii) (c) Cube (vi) (f) Cone
Match the shape with the name:
2024-25
VISUALISING SOLID SHAPES 199
Try to identify some objects shaped like each of these.
By a similar argument, we can say figures drawn on paper which have only length and
breadth are called two dimensional (i.e., plane) figures. We have also seen some two
dimensional figures in the earlier classes.
Match the 2 dimensional figures with the names (Fig 13.2):
(i) (a) Circle
(ii) (b) Rectangle
(iii) (c) Square
(iv) (d) Quadrilateral
(v) (e) Triangle
Fig 13.2
Note: We can write 2-D in short for 2-dimension and 3-D in short for
3-dimension.
13.2  FACES, EDGES AND VERTICES
Do you remember the Faces, Vertices and Edges of solid shapes, which you studied
earlier?  Here you see them for a cube:
         (i)            (ii)           (iii)
Fig 13.3
The 8 corners of the cube are its vertices. The 12 line segments that form the
skeleton of the cube are its edges. The 6 flat square surfaces that are the skin of the
cube are its faces.
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MATHEMATICS 200
Can you see that, the two dimensional figures can be identified as the faces of the
three dimensional shapes?  For example a cylinder  has two faces which are circles,
and a pyramid, shaped like this  has triangles as its faces.
We will now try to see how some of these 3-D shapes can be visualised on a 2-D
surface, that is, on paper.
In order to do this, we would like to get familiar with three dimensional objects closely .
Let us try forming these objects by making what are called nets.
13.3  NETS FOR BUILDING 3-D SHAPES
T ake a cardboard box.  Cut the edges to lay the box flat.  Y ou have now a net for that box.
A net is a sort of skeleton-outline in 2-D [Fig13.4 (i)], which, when folded [Fig13.4 (ii)],
results in a 3-D shape [Fig13.4 (iii)].
DO THIS
Vertex
Face
Edge
Face
Vertex
Edge
Faces (F) 6 4
Edges (E) 12
V ertices (V) 8 4
Complete the following table:
Table 13.1
(i) (ii) (iii)
Fig 13.4
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VISUALISING SOLID SHAPES 201
Here you got a net by suitably separating the edges. Is the
reverse process possible?
Here is a net pattern for a box (Fig 13.5). Copy an enlarged
version of the net and try to make the box by suitably folding
and gluing together. (Y ou may use suitable units). The box is a
solid. It is a 3-D object with the shape of a cuboid.
Similarly, you can get a net for a cone by cutting a slit along
its slant surface (Fig 13.6).
You have different nets for different shapes. Copy
enlarged versions of the nets given (Fig 13.7) and try to make
the 3-D shapes indicated. (Y ou may also like to prepare skeleton
models using strips of cardboard fastened with paper clips).
Fig 13.5
Cube
(i)
Cone
(iii)
Cylinder
(ii)
Fig 13.6
W e could also try to make a net for making a pyramid like the Great Pyramid in Giza
(Egypt) (Fig 13.8). That pyramid has a square base and triangles on the four sides.
Fig 13.7
Fig 13.9
Fig 13.8
See if you can make it with the given net (Fig 13.9).
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Page 5


MATHEMATICS 198
Visualising Solid
Shapes
Chapter  13
13.1  INTRODUCTION: PLANE FIGURES AND SOLID SHAPES
In this chapter, you will classify figures you have seen in terms of what is known as
dimension.
In our day to day life, we see several objects like books, balls, ice-cream cones etc.,
around us which have different shapes. One thing common about most of these objects is
that they all have some length, breadth and height or depth.
That is, they all occupy space and have three dimensions.
Hence, they are called three dimensional shapes.
Do you remember some of the three dimensional shapes (i.e., solid shapes) we have
seen in earlier classes?
TRY THESE
Fig 13.1
(i) (a) Cuboid (iv) (d) Sphere
(ii) (b) Cylinder (v) (e) Pyramid
(iii) (c) Cube (vi) (f) Cone
Match the shape with the name:
2024-25
VISUALISING SOLID SHAPES 199
Try to identify some objects shaped like each of these.
By a similar argument, we can say figures drawn on paper which have only length and
breadth are called two dimensional (i.e., plane) figures. We have also seen some two
dimensional figures in the earlier classes.
Match the 2 dimensional figures with the names (Fig 13.2):
(i) (a) Circle
(ii) (b) Rectangle
(iii) (c) Square
(iv) (d) Quadrilateral
(v) (e) Triangle
Fig 13.2
Note: We can write 2-D in short for 2-dimension and 3-D in short for
3-dimension.
13.2  FACES, EDGES AND VERTICES
Do you remember the Faces, Vertices and Edges of solid shapes, which you studied
earlier?  Here you see them for a cube:
         (i)            (ii)           (iii)
Fig 13.3
The 8 corners of the cube are its vertices. The 12 line segments that form the
skeleton of the cube are its edges. The 6 flat square surfaces that are the skin of the
cube are its faces.
2024-25
MATHEMATICS 200
Can you see that, the two dimensional figures can be identified as the faces of the
three dimensional shapes?  For example a cylinder  has two faces which are circles,
and a pyramid, shaped like this  has triangles as its faces.
We will now try to see how some of these 3-D shapes can be visualised on a 2-D
surface, that is, on paper.
In order to do this, we would like to get familiar with three dimensional objects closely .
Let us try forming these objects by making what are called nets.
13.3  NETS FOR BUILDING 3-D SHAPES
T ake a cardboard box.  Cut the edges to lay the box flat.  Y ou have now a net for that box.
A net is a sort of skeleton-outline in 2-D [Fig13.4 (i)], which, when folded [Fig13.4 (ii)],
results in a 3-D shape [Fig13.4 (iii)].
DO THIS
Vertex
Face
Edge
Face
Vertex
Edge
Faces (F) 6 4
Edges (E) 12
V ertices (V) 8 4
Complete the following table:
Table 13.1
(i) (ii) (iii)
Fig 13.4
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VISUALISING SOLID SHAPES 201
Here you got a net by suitably separating the edges. Is the
reverse process possible?
Here is a net pattern for a box (Fig 13.5). Copy an enlarged
version of the net and try to make the box by suitably folding
and gluing together. (Y ou may use suitable units). The box is a
solid. It is a 3-D object with the shape of a cuboid.
Similarly, you can get a net for a cone by cutting a slit along
its slant surface (Fig 13.6).
You have different nets for different shapes. Copy
enlarged versions of the nets given (Fig 13.7) and try to make
the 3-D shapes indicated. (Y ou may also like to prepare skeleton
models using strips of cardboard fastened with paper clips).
Fig 13.5
Cube
(i)
Cone
(iii)
Cylinder
(ii)
Fig 13.6
W e could also try to make a net for making a pyramid like the Great Pyramid in Giza
(Egypt) (Fig 13.8). That pyramid has a square base and triangles on the four sides.
Fig 13.7
Fig 13.9
Fig 13.8
See if you can make it with the given net (Fig 13.9).
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MATHEMATICS 202
EXERCISE 13.1
1. Identify the nets which can be used to make cubes (cut out copies of the nets and try it):
(i) (ii) (iii)
(iv) (v) (vi)
2. Dice are cubes with dots on each face.  Opposite faces of a die always have a total
of seven dots on them.
Here are two nets to make dice (cubes); the numbers inserted in each square indicate
the number of dots in that box.
Insert suitable numbers in the blanks, remembering that the number
on the opposite faces should total to 7.
3. Can this be a net for a die?
Explain your answer .
TRY THESE
Here you find four nets (Fig 13.10). There are two correct nets among them to make
a tetrahedron.  See if you can work out which nets will make a tetrahedron.
Fig 13.10
1    2
3     4
5    6
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