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Surface Areas and Volumes

Look at the following figure.

In day to day life, we come across various shapes similar to this, such as a glass, a bucket, a funnel, etc. This shape is known as frustum of cone.

Now, can we describe what a frustum is?

The solid obtained by cutting a cone by a plane parallel to its base is known as frustum of a cone.

We can observe from the following figure that a cone is sliced by a plane parallel to the base. After removing the smaller cone, we obtain a solid which is known as frustum of cone.

The given figure shows an iron tub and its dimensions. Its exterior curved surface is to be polished. Can we find the area of iron tub to be polished?

The given video will explain the application of formula for curved surface area of a frustum.

To find the total surface area of a frustum, we just need to add the areas of its two circular faces to its curved surfaces area.

Therefore,

Total surface area of frustum = Curved surface area + Area of two circular faces

Total surface area of a frustum = π(r1 + r2)l + πr12 + πr22

Let us look at some examples to understand the concept better.

Example 1:

A shuttle cock, which is used for playing badminton, has the shape of a frustum mounted on a hemisphere as shown in the following figure. The dimensions are also shown in the figure. Find the outer curved surface area of the shuttle cock. (Use π = 3.14)

Solution:

It is given that,

r1 = radius of upper end of frustum = 3.5 cm

r2 = radius of lower end of frustum = 1.5 cm (Also the radius of the hemisphere)

h = height of the frustum = cm

Now, slant height,

= 6.32 cm

Outer curved surface area of shuttle cock = C.S.A. of frustum + C.S.A. of hemisphere

= +

=

= 99.22 + 14.13

= 113.35 cm2

Thus, the required surface area is 113.35 cm2.

Example 2:

A solid is in the shape of the frustum of a cone. The radii of its circular ends are 7 cm and 14 cm. The total surface area of the solid is 2420 cm2. What is the height of the solid?

Solution:

The solid is in the shape of a conical frustum with radii 7 cm and 14 cm and total surface area 2420 cm2.

Let r1 and r2 be the radii of the solid. Let l and h be its slant height and height respectively.

The total surface area of the frustum is given by .

Here, r1 = 14 cm and r2 = 7 cm

Now,

Thus, the height of the solid is 24 cm.

Example 3:

The given fi...

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