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About This Booklet on Surface Areas and Volumes

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About This Booklet on Surface Areas and Volumes

This Particular Booklet has been designed for the students of Math Class 10 (CBSE Board).

However, it will also help those who have these chapters in their curriculum or want to gain the

knowledge and explore the concepts.

This Booklet explains:

1. Cube, Cuboid and Cylinder

2. Cone and Frustum

3. Sphere and Hemisphere

4. Combination of solids

This Booklet also covers:

1. MCQs

2. Questions with Solutions

3. Questions for practice and

4. QR Codes to scan and watch videos on Surface Areas and Volumes

QR codes when scanned with mobile scanner take you to our YouTube channel Let’sTute

(www.youtube.com/letstute) where you can watch our free videos (need internet connection)

on the topic.

For Surface Areas and Volumes, in total, we have 8 videos which are accessible on several other

platforms as described on the back cover of this Booklet.

However, if there is any query, feel free to connect with us on the details given on the last page.

Some other documents are also available:

About Let’s Tute Let’sTute (Universal Learning Aid) is an E-learning company based in Mumbai, India.

(www.letstute.com) We create content for Mathematics, Biology, Physics, Environmental

Science, Book-Keeping & Accountancy and also a series on value education known as V2lead

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1. A cube is special type of cuboid.

A)True B)False

2. Volume of any object is nothing but its maximum capacity of storage.

A)True B)False

3. A cube is three dimensional objects bounded by _ _ _ _ equal square faces.

A)Five B)Six C)Four D) None of these

MCQs (Multiple Choice Questions)

Scan to watch the video on

Surface area and Volume

13.1 Cube, Cuboid and Cylinder

Scan to watch the video on

Volume of cube, cuboid & cylinder

Surface Area: It is the sum of total exposed area of a three dimensional solid object. Its unit

can be in the form cm2, m2 etc.

Volume: It is the amount of space occupied by an object. Its unit can be cm3, m3, etc.

Cube: A cube is a special form of cuboid which is bounded by six equal square faces.

Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its

sides, is called as Right Circular Cylinder.

Cuboid: A cuboid is the solid shape which has six rectangle faces at right angles to each other.

Total Surface Area and Volume for Cube, Cuboid and Cylinder

INTRODUCTION

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4. A cube of sides 6 m will have volume_ _ _ _

A)216 m2 B) 216 m3 C) 216 cm2 D) 216 m

5. If the radius of cylinder is doubled then the curved surface area of cylinder will be _ _

_ _ as compared to previous.

A)Same B) Doubled C) Thrice D) None of these

6. What is the surface area of a cube having edges of length 15 cm?

A)1350 cm B)225 cm C)1350 cm D) 1500 cm

7. What is the length of side of a cube if the total surface area of cube is 726 cm2?

A)11 cm B)16 cm C)72 cm D) None of these

8. If the area of one face of cube is 169 cm2 then what will be the volume of cube.

A) 1300 cm B) 4238 cm C) 5642 cm D) 2197 cm

9. If the perimeter of one face of the cube is 28 cm then what will be its volume.

A) 343 cm B) 343 cm C) 437 cm D) 473 cm

10. What is the surface area of cuboid with dimension in m are 12×10×15

A) 1200 m B) 900 m C) 920 m D) 940 m

11. Total Surface Area of Right Circular Cylinder is _ _ _ _

A) 𝜋 ℎ + B) 𝜋 ℎ × C) 𝜋 ℎ − D) None of these

12. What is the curved surface area of Cylinder when radius is 14 cm and height is 15 cm.

[ π = 22/7]

A) 1300 B) 2100 C) 1320 D) 4242

13. How many liters of water will be stored in a cylindrical tank of radius 3 m and height

7 m. [ Hint 1 m3 =1000 Liters take π = 22/7]

A) 28640 𝐿 B) 254000 𝐿 C) 198000 𝐿 D) None of these

Q.1 Find the Total cost of painting the four walls of cuboidal room at the rate of 13 Rs per

m2.The Dimension of the Cuboidal room are of length 10m , breadth 5m and height 6m.

Given: For Cuboidal Room,

Rate of Painting = 13 Rs/m2, l = 10m, b = 5m, h = 6m

Solution:

Answers:

1.(A), 2.(A), 3.(B), 4.(B), 5.(B), 6.(C), 7.(A), 8.(D), 9.(A) 10.(B) 11.(A) 12.(C) 13.(C)

Solved Questions

An Approach: As we have to paint only the walls, surface area of roof and floor so are not

taken into calculation

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Vertical surface area of cuboid (four walls) = + × 𝒉

Vertical surface area of cuboid = 180 m2.

Total cost of painting four walls =

Vertical surface area of cuboid × Rate of painting wall/m2

= 180 m2 × 13

= 2340

Ans. Therefore Cost of Painting the four walls is Rs 2340

Q.2 A cuboidal pillar of dimension 600 cm × 55 cm × 25 cm made of wood which weights

200kg/m3. What will be the weight of the pillar?

Given:

For Pillar [cuboid]

Length = 600cm = = 6m

Breadth = 55cm = = 0.55m

Height = 25cm = = 0.25m

Solution:

Volume of pillar [cuboid] = × × 𝒉

= 6 m × 0.55 m× 0.25 m

= 0.825 m3

Weight of pillar = Volume of pillar × weight of volume per m3

= 0.825×200

= 165 Kg

Ans. Weight of pillar =165 Kg

Q.3 If the Diameter of a roller is 40 cm and length is 140 cm. It takes 350 revolutions to land

a playground. Find the area of playground in m2.

Given: The Diameter of cylindrical roller = 40 cm =0.4m

Radius of cylindrical roller(r) = = 20 cm= 0.2 m

Length of cylindrical roller (h) =140 cm = 1.4 m

Solution:

Curved surface area of cylindrical roller = 𝝅𝒓𝒉

= × × . × .

= × × . × .

= 1.76 m2

Area covered in one revolution is = curved surface area of cylinder

= 1.76 m2

l b

h = 2(10+5) × 6

= 2(15) × 6

= 180

Tips: Please remember to convert

dimensions from centimeter to meter as

weight is given kg/m3.

An Approach: Please try to analyze

that 1 revolution = lateral surface area

of cylinder. So 350 revolution = × 𝜋 ℎ

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Total Area of Playground = No. of revolutions × curved surface area of cylinder

= 350 × 1.76 m2

= 616 m2

Ans. Area of Playground = 616 m2

Q.4 The curved surface area of cylinder is 154 cm2. The total surface area of the cylinder is

thrice the curved surface area. Find the volume of cylinder. [𝛑 = ] Given: Curved surface area of cylinder ( 𝜋 ℎ ) = 154 cm2

Solution: Total surface area of cylinder = 3 × curved surface area

𝜋 ℎ + 𝜋 = ×

154 + 2πr2 = 462

2πr2 = 462 – 154

2πr2 = 308

× × =

r2= ××

r2= 49

r = 7

Curved surface area of cylinder ( 𝝅𝒓𝒉 ) = 154 cm2

× × × ℎ =

h = ×

h = 3.5 cm

Volume of Cylinder = 𝝅𝒓 𝒉

= × × × .

= 539 cm3

Ans. Volume of Cylinder = 539 cm3

Q1. Find the length of edges of cube having volume 512 cm3.

Ans: 8 cm

Q2. A Gift box of length 15 cm, breadth 13cm and height 14 cm need to be covered with a

decorative wrapper. Find the area of wrapper needed.

Ans: 1174 cm2

Q3. If a gas cylinder has an inner radius of 14 cm and height of 70 cm. How much maximum

amount of gas can be stored in that cylinder? [π = 22/7]

Ans: 43120 cm3

Practice Yourself

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Q4.What will be the radius of cylinder if the curved surface area of cylinder is 1760 cm2 and its

height is 14 cm?

Ans: 20 cm

Q5. If curved surface area of cylinder is 1188 cm2 and radius is 21 cm. What will be its volume?

[Hint: first find the height using curved surface area]

Ans: 12474 cm3

Q6. Find the curved surface area of cylinder when radius is 14 cm and height is 2 cm more than

double of its radius.

Ans: 2640 cm2

Make your own notes:

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1. If the radii of both the ends of frustum are same then it is a cylinder.

A) True B) False

2. The volume of a cone is one third of the volume of the cylinder.

A) True B) False

3. What is surface area of a cone with radius 7 cm and slant height 12 cm?

A) 184 cm2 B) 418 cm2 C) 184 cm2 D) 284 cm2

4. Curved surface area of a cone with radius 14 cm and slant height 10 cm …. A) 400 cm2 B) 404 cm2 C) 440 cm2 D) 284 cm2

MCQs (Multiple Choice Questions)

INTRODUCTION

Cone: A cone is a three-dimensional geometric shape that tapers smoothly from a flat circular

base to a point called vertex.

We can also define it as, “A solid obtained by revolving a right angled triangular lamina about

any side (except hypotenuse) is a right circular cone.”

Frustum: If the cone is cut off by a plane parallel to the base not passing through vertex then we get

the lower base portion as a frustum of cone.

13.2 Cone and Frustum

l = slant height, r = radius of cone

Slant Height of a Cone (l) = 𝒉 + 𝒓

Curved surface area of Cone = 𝝅𝒓𝒍 Total Surface Area of Cone = 𝝅𝒓 𝒓+ 𝒍

Volume of Cone = 𝝅𝒓 𝒉

Slant Height of the Frustum (l) = 𝒉 + 𝒓 − 𝒓

Curved surface area of Frustum = 𝝅 𝒓 + 𝒓 𝒍 Total Surface Area of Frustum = 𝝅 𝒓 + 𝒓 𝒍+ 𝝅𝒓 + 𝝅𝒓

Volume of Frustum = 𝝅 𝒓 + 𝒓 + 𝒓 × 𝒓 𝒉

Where r1 and r2 are the

radii of the ends of

frustum (r1>r2) h = height

and l = slant height

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5. Frustum can be obtained by cutting _ _

A) Sphere B) Cylinder C) Cone D) Cube

6. Slant height of a cone with radius 3 cm and height 4 cm…

A) 3.5 cm B) 4 cm C) 7 cm D) 5 cm

7. Total surface area of frustum of a cone is ….. A) 𝜋 − + 𝜋 + 𝜋 B) 𝜋 + + 𝜋 + 𝜋

C) 𝜋 − + 𝜋 − 𝜋 D) None of these

Q.1 The height of a cone is 4 cm and radius is 3 cm. what will be its Total surface area and

volume?

Given: Radius of cone (r) = 3 cm

Height of cone (h) = 4 cm

Solution:

Slant height of cone (l) = 𝒉 + 𝒓 = √ + = √ + = √

l = 5 cm

Total Surface Area of Cone = 𝝅𝒓 𝒓 +

= × × +

= × ×

= 75.428 cm2

Volume of cone = 𝝅𝒓 𝒉

= × × ×

= × ×

= 37.714 cm3

Ans. Total Surface Area of Cone=75.428 cm2, Volume of cone=37.714 cm3

Q.2 The volume of the cone is 352 cm3 and its height is 21 cm. what will be the radius of

cone?

Given:

Volume of cone (v) = 352 cm3

Height of cone = 21 cm

Answers: 1.(A), 2.(A), 3.(B), 4.(C), 5.(C), 6.(D), 7.(B)

Solved Questions

An Approach: First we will find the slant height

of the cone (l)

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Solution:

Volume of cone (v) = 𝝅𝒓 𝒉

352 = × × 𝒓 ×

=

=

∴ = 𝑖 𝑔 ℎ 𝑖

Ans. The radius of cone is 4 cm.

Q.3 The height of frustum is 4 cm and its radii are 6 cm and 3 cm. Find curved surface area

and volume of frustum.

Given:

Height of frustum (h) = 4 cm

Base Radius (r1) = 6 cm

Upper Radius (r2) =3 cm

Solution:

Slant Height of the Frustum (l ) = 𝒉 + 𝒓 − 𝒓

l = + −

l = √ +

l = √

l =5 cm

Curved surface area of frustum = 𝝅 𝒓 + 𝒓

= × + ×

= × 𝟗 ×

= 𝟗𝟗

=141.42 cm2

Volume of Frustum = 𝝅 𝒓 + 𝒓 + 𝒓 × 𝒓 𝒉

= × + + × ×

= × × + + ×

= × ×

= 264 cm3

Ans. Curved surface area of frustum is 141.42 , Volume of frustum is 264

Q.4 The curved surface area of frustum of a cone is 130 cm2 and the circumferences of the

circular bases are 16 cm and 10 cm respectively. What will be slant height of frustum of cone?

Given:

Curved surface area of frustum of a cone = 130

Circumference of base circle = 16 cm

Circumference of upper circle = 10 cm

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Solution:

Circumference of base circle = 2πr1

16 = 2πr1

𝜋 =

= 𝜋

Circumference of upper circle = 2πr2

10 = 2πr2

𝜋 =

= 𝝅

Curved surface area of frustum = 𝝅 𝒓 + 𝒓

130 = 𝜋 × 𝜋 + 𝜋 ×

130 = 13 ×

=

=

Ans: Slant height of frustum of cone is 10 cm

Q.1.If the radius of a cone is 5 cm and height is 12 cm. What is its slant height and total surface

area?

Ans: 13 cm and 283.86 cm2

Q.2. If the diameter of the base of a cone is 6 cm and slant height is 14 cm .Find the curved

surface area of a cone?

Ans: 132 cm2

Q.3. A bucket is in the shape of the frustum of a right circular cone, its height is 4 cm and the

bases of radii are 7 cm and 4 cm. Find the volume and the total surface area of the bucket.

Ans: 389.71 cm3, 377.135 cm2

Q.4. Find the volume of a frustum of a cone with bases of radii 7cm and 5cm. The height of this

frustum is 8cm.

Ans: 913.52 cm3

Practice Yourself

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1. The volume of sphere is twice the volume of hemisphere.

A) True B) False

2. The total surface areas of solid and hollow hemisphere are same.

A) True B) False

3. If the radius of cylinder and sphere is same then lateral surface area of cylinder equals

surface area of sphere.

A) True B) False

MCQs (Multiple Choice Questions)

Sphere: A solid obtained by revolving a circular lamina

about any of its diameter is called a sphere.

We can also define it as, “The set of all points in space

which are equidistant from a fixed point.”

Hemisphere: A plane passing through the center of a sphere divides sphere into two equal parts.

Each part is called a hemisphere.

13.3 Sphere and Hemisphere

Surface Area of Sphere = 𝝅𝒓

Volume of Sphere = 𝝅𝒓

Where ‘r’ = radius of sphere

A B A B

B

A

Scan to watch the video on

Surface Area of Sphere

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4. The surface area of a sphere is - - - - - times the area of a circle

A) Three B) Four C) Five D) None of these

5. If the volume of a sphere is 540 cm3, volume of hemisphere will be ….. A) 270 cm3 B) 250 cm3 C) 200 cm3 D) 220 cm3

6. Curved surface area of hemisphere with radius 14 cm is …. A) 1232 cm2 B) 1323 cm2 C) 1223 cm2 D) 1332 cm2

7. Volume of sphere with radius 3.5 cm is

A) 97 cm3 B) 77.17 π cm3 C) 57.17 π cm3 D) 89.67 cm3

8. What is the surface area of a sphere with radius 7 cm? 𝛑 =

A) 166 cm2 B) 661 cm2 C) 616 cm2 D) None of these

9. Volume of hemisphere is …… if the radius is 2.1 cm

A) 9.21π cm3 B) 5π cm3 C) 16.44 π cm3 D) 6.174 π cm3

10. What will be the total surface area of solid hemisphere if the area of base circle is 250

cm2 ….. A) 750 cm2 B) 550 cm2 C) 1000 cm2 D) 700 cm2

Q.1 The surface area of sphere is 113.04 cm2.Find the radius and volume?

(π = 3.14)

Given:

Area of Sphere = 113.04

Solution:

Let the radius of sphere = r

Surface Area of sphere = 𝜋

113.04 = 4×3.14×

.× . = × =

9 =

r = 3 cm

Volume of sphere = 𝝅𝒓 = × . ×

= × . ×

= 113.04 cm3

Ans. Radius of Sphere (r) = 3 cm Volume of Sphere = 113.04 cm3

Answers:

1.(A), 2.(B), 3.(A), 4.(B), 5.(A), 6.(A), 7.(C), 8.(C) 9.(D) 10.(A)

Solved Questions

An Approach: To get the volume, we need radius. So

even if the question doesn’t ask us to find radius, we will have to find it to help us get the volume.

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Q.2 The volume of the sphere is 288π cm3. Find its surface area and radius.

Given:

Volume of Sphere (v) = 288π cm3

Solution:

Volume of Sphere = πr

288𝜋 = 𝜋

×

=

216 =

∴ = 𝑖 𝑔 ℎ 𝑖

Surface Area of Sphere = 𝝅𝒓

= 4 × 3.14 ×

= 452.16 cm2

Ans: The radius of sphere is 6 cm. Surface Area of sphere is 452.16 cm2

Q.3 If volume of two spheres is in the ratio 125:216, find their radii if the sum of their radii

is 22 cm.

Given:

Ratio of Volumes of two Spheres is 125:216 and sum of their radii is 22 cm.

Solution:

Let r1 and r2 be the radii and v1 and v2 be the volumes of two spheres respectively.

Volume of Sphere = πr

∴ 𝑣𝑣 = 𝜋𝑟𝜋𝑟

∴ = 𝑟𝑟

∴ = 𝑟𝑟 (Taking cube root)

∴ = -------(1)

+ = ---------(2) (Given)

+ = (From 1)

𝑟 + 𝑟 =

+ = ×

=

=

∴ =

Substitute value of in equation (2)

+ =

= −

∴ =

Ans. Radii of spheres are 10 cm and 12 cm

An Approach: We need radius first to get the surface

area. So we will equate it with the formula of volume

to find the radius and then the surface area.

An Approach: Here ratio of volumes and sum of

radii is given hence we’ll get two equation to get the two unknown radii.

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Q.1. If the radius of sphere is 7 cm. What is its volume and surface area?

Ans: 1437.33 cm3 and 616 cm2

Q.2. If the diameter of the base circle of solid hemisphere is 2.8 cm .Find the curved surface area

of hemisphere?

Ans: 12.32 cm2

Q.3. If the radius of hemisphere is doubled what will be the ratio of its total surface area to that

of original one?

Ans: 4:1

Q.4. Find the volume and curved surface area of a hemisphere with radius 4.2 cm

(π = )

Ans: 155.232 cm3, 110.88 cm2

Q.5. A solid metal sphere 16 cm in diameter is melted and recast into smaller solid metal spheres

4 cm in diameter. How many of these smaller solid metal spheres will there be?

(Hint: When we molt, surface area may change of newly formed solid but volume remains same.

So Volume of sphere (R = 8m) = Number of smaller sphere × Volume of sphere (r = 2 cm).

Ans: 64

Make your own notes:

Practice Yourself

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