SURFACE AREAS AND VOLUMES 161
SURFACE AREAS AND
VOLUMES 12
12.1 Introduction
From Class IX, you are familiar with some of the solids like cuboid, cone, cylinder, and
sphere (see Fig. 12.1). You have also learnt how to find their surface areas and volumes.
Fig. 12.1
In our day-to-day life, we come across a number of solids made up of combinations
of two or more of the basic solids as shown above.
You must have seen a truck with a
container fitted on its back (see Fig. 12.2),
carrying oil or water from one place to
another. Is it in the shape of any of the four
basic solids mentioned above? You may
guess that it is made of a cylinder with two
hemispheres as its ends.
Fig. 12.2
Rationalised 2023-24
, 162 MATHEMATICS
Again, you may have seen an object like the
one in Fig. 12.3. Can you name it? A test tube, right!
You would have used one in your science laboratory.
This tube is also a combination of a cylinder and a
hemisphere. Similarly, while travelling, you may have
seen some big and beautiful buildings or monuments
made up of a combination of solids mentioned above.
If for some reason you wanted to find the
surface areas, or volumes, or capacities of such
objects, how would you do it? We cannot classify
these under any of the solids you have already studied. Fig. 12.3
In this chapter, you will see how to find surface areas and volumes of such
objects.
12.2 Surface Area of a Combination of Solids
Let us consider the container seen in Fig. 12.2. How do we find the surface area of
such a solid? Now, whenever we come across a new problem, we first try to see, if
we can break it down into smaller problems, we have earlier solved. We can see that
this solid is made up of a cylinder with two hemispheres stuck at either end. It would
look like what we have in Fig. 12.4, after we put the pieces all together.
Fig. 12.4
If we consider the surface of the newly formed object, we would be able to see
only the curved surfaces of the two hemispheres and the curved surface of the cylinder.
So, the total surface area of the new solid is the sum of the curved surface
areas of each of the individual parts. This gives,
TSA of new solid = CSA of one hemisphere + CSA of cylinder
+ CSA of other hemisphere
where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’
respectively.
Let us now consider another situation. Suppose we are making a toy by putting
together a hemisphere and a cone. Let us see the steps that we would be going
through.
Rationalised 2023-24
SURFACE AREAS AND
VOLUMES 12
12.1 Introduction
From Class IX, you are familiar with some of the solids like cuboid, cone, cylinder, and
sphere (see Fig. 12.1). You have also learnt how to find their surface areas and volumes.
Fig. 12.1
In our day-to-day life, we come across a number of solids made up of combinations
of two or more of the basic solids as shown above.
You must have seen a truck with a
container fitted on its back (see Fig. 12.2),
carrying oil or water from one place to
another. Is it in the shape of any of the four
basic solids mentioned above? You may
guess that it is made of a cylinder with two
hemispheres as its ends.
Fig. 12.2
Rationalised 2023-24
, 162 MATHEMATICS
Again, you may have seen an object like the
one in Fig. 12.3. Can you name it? A test tube, right!
You would have used one in your science laboratory.
This tube is also a combination of a cylinder and a
hemisphere. Similarly, while travelling, you may have
seen some big and beautiful buildings or monuments
made up of a combination of solids mentioned above.
If for some reason you wanted to find the
surface areas, or volumes, or capacities of such
objects, how would you do it? We cannot classify
these under any of the solids you have already studied. Fig. 12.3
In this chapter, you will see how to find surface areas and volumes of such
objects.
12.2 Surface Area of a Combination of Solids
Let us consider the container seen in Fig. 12.2. How do we find the surface area of
such a solid? Now, whenever we come across a new problem, we first try to see, if
we can break it down into smaller problems, we have earlier solved. We can see that
this solid is made up of a cylinder with two hemispheres stuck at either end. It would
look like what we have in Fig. 12.4, after we put the pieces all together.
Fig. 12.4
If we consider the surface of the newly formed object, we would be able to see
only the curved surfaces of the two hemispheres and the curved surface of the cylinder.
So, the total surface area of the new solid is the sum of the curved surface
areas of each of the individual parts. This gives,
TSA of new solid = CSA of one hemisphere + CSA of cylinder
+ CSA of other hemisphere
where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’
respectively.
Let us now consider another situation. Suppose we are making a toy by putting
together a hemisphere and a cone. Let us see the steps that we would be going
through.
Rationalised 2023-24
