**Chapter 13 Surface Areas and Volumes**

• 13.1 Introduction

• 13.2 Surface Area of a Combination of Solids

•13.3 Volume of a Combination of Solids

• 13.4 Conversion of Solid from One Shape to Another

• 13.5 Frustum of a Cone

• 13.6 Summary

Maths NCERT solutions class 10 Chapter 13 Surface Areas and Volumes

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**Important formula**

*Exercise 13.1*

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Unless stated otherwise, take π = 22/7

Q.1 2 cubes each of volume 64 cm3

are joined end to end. Find the surface area of the

resulting cuboid.

2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The

diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the

inner surface area of the vessel.

3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius.

The total height of the toy is 15.5 cm. Find the total surface area of the toy.

4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest

diameter the hemisphere can have? Find the surface area of the solid.

5. A hemispherical depression is cut out from one face of a cubical wooden block such

that the diameter l of the hemisphere is equal to the edge of the cube. Determine the

surface area of the remaining solid.

6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 13.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and

diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the

top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of

the canvas of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will not

be covered with canvas.)

8. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the

same height and same diameter is hollowed out. Find the total surface area of the

remaining solid to the nearest cm2.

9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 13.11. If the height of the cylinder is 10 cm, and its

base is of radius 3.5 cm, find the total surface area of the article.

*Exercise 13.2*

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1 . A solid is in the shape of a cone standing on a hemisphere with both their radii being

equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid

in terms of π.

2. Rachel, an engineering student, was asked to make a model shaped like a cylinder with

two cones attached at its two ends by using a thin aluminium sheet. The diameter of the

model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume

of air contained in the model that Rachel made. (Assume the outer and inner dimensions

of the model to be nearly the same.)

3. A gulab jamun, contains sugar syrup up to about

30% of its volume. Find approximately how much

syrup would be found in 45 gulab jamuns, each

shaped like a cylinder with two hemispherical ends

with length 5 cm and diameter 2.8 cm (see Fig. 13.15).

4. A pen stand made of wood is in the shape of a

cuboid with four conical depressions to hold pens.

The dimensions of the cuboid are 15 cm by 10 cm by

3.5 cm. The radius of each of the depressions is 0.5

cm and the depth is 1.4 cm. Find the volume of

wood in the entire stand (see Fig. 13.16).

5. A vessel is in the form of an inverted cone. Its

height is 8 cm and the radius of its top, which is

open, is 5 cm. It is filled with water up to the brim.

When lead shots, each of which is a sphere of radius

0.5 cm are dropped into the vessel, one-fourth of

the water flows out. Find the number of lead shots

dropped in the vessel.

6. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which

is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the

pole, given that 1 cm3

of iron has approximately 8g mass. (Use π = 3.14)

7. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on

a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water

such that it touches the bottom. Find the volume of water left in the cylinder, if the radius

of the cylinder is 60 cm and its height is 180 cm.

8. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter

of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its

volume to be 345 cm3

. Check whether she is correct, taking the above as the inside

measurements, and π = 3.14.

*Exercise 13.3*

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Q.1 A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of

radius 6 cm. Find the height of the cylinder.

2. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single

solid sphere. Find the radius of the resulting sphere.

3. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out

to form a platform 22 m by 14 m. Find the height of the platform.

4. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly

all around it in the shape of a circular ring of width 4 m to form an embankment. Find the

height of the embankment.

5. A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm

is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter

6 cm, having a hemispherical shape on the top. Find the number of such cones which can

be filled with ice cream.

6. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form

a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?

7. A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This

bucket is emptied on the ground and a conical heap of sand is formed. If the height of the

conical heap is 24 cm, find the radius and slant height of the heap.

8. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much

area will it irrigate in 30 minutes, if 8 cm of standing water is needed?

9. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in

her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the

rate of 3 km/h, in how much time will the tank be filled?

*Exercise 13.4*

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Q. 1 A drinking glass is in the shape of a frustum of a

cone of height 14 cm. The diameters of its two

circular ends are 4 cm and 2 cm. Find the capacity of

the glass.

2. The slant height of a frustum of a cone is 4 cm and

the perimeters (circumference) of its circular ends

are 18 cm and 6 cm. Find the curved surface area of

the frustum.

3. A fez, the cap used by the Turks, is shaped like the

frustum of a cone (see Fig. 13.24). If its radius on the

open side is 10 cm, radius at the upper base is 4 cm

and its slant height is 15 cm, find the area of material

used for making it.

4. A container, opened from the top and made up of a metal sheet, is in the form of a

frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20

cm, respectively. Find the cost of the milk which can completely fill the container, at the

rate of 20 per litre. Also find the cost of metal sheet used to make the container, if it costs 8 per 100 cm2

. (Take π = 3.14)

5. A metallic right circular cone 20 cm high and whose vertical angle is 60° is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1cm, 16 find the length of the wire.