# NCERT Solutions for Class 9 Maths Chapter 13 Surface Area and Volumes

NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes: It is among one of the best topics which has so much use in real-life problems. The concepts of this chapter are used in our daily life for calculations like- how many liters of paint is needed to paint an entire house? How many boxes are needed to fill these chocolates? what should be the area of the sheet I should by to cover the roof of my house? These types of calculations can be done using the concept of surface areas and volumes. CBSE NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes are covering every problem in a very detailed manner. In this chapter, you will study the shapes- cube, cuboid, cylinder, cone, sphere, hemisphere, etc. You have to face the questions related to the curved surface area, total surface area, volumes, and many others.

Solutions of NCERT for class 9 maths chapter 13 Surface Area and Volumes are designed to provide you assistance while solving the practice exercises. The language of the questions sometimes is not direct, you have to identify that what the question is specifically about. There are a total of 9 exercises consisted of a total of 102 questions including the optional exercise. NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes have the exam-oriented solutions to all the questions of practice exercises. NCERT solutions are available for free which you can get by clicking on the given link.

## NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.1

Given, dimensions of the plastic box

Length,

Width,

Depth,

(i) The area of the sheet required for making the box (open at the top)= Lateral surface area of the box. + Area of the base.

=

The required area of the sheet required for making the box is

Given, dimensions of the plastic box

Length,

Width,

Depth,

We know, area of the sheet required for making the box is

(ii) Cost for   of sheet = Rs 20

Cost for  of sheet =

Required cost of the sheet is

Given,

Dimensions of the room =

Required area to be whitewashed = Area of the walls + Area of the ceiling

Cost of white-washing per  area =
Cost of white-washing area =

Therefore, the required cost of whitewashing the walls of the room and the ceiling is

Given,

The perimeter of rectangular hall =

Cost of painting the four walls at the rate of Rs 10 per  = Rs 15000

Let the height of the wall be

Area to be painted =

Required cost =

Therefore, the height of the hall is

Given, dimensions of the brick =

We know, Surface area of a cuboid =

The surface area of a single brick =

Number of bricks that can be painted =

Therefore, the required number of bricks that can be painted = 100

Which box has the greater lateral surface area and by how much?

Given,

Edge of the cubical box =

Dimensions of the cuboid =

The lateral surface area of the cubical box =

The lateral surface area of the cuboidal box =

Clearly, Lateral surface area of the cubical box is greater than the cuboidal box.

Difference between them =

Which box has the smaller total surface area and by how much?

Given,

Edge of the cubical box =

Dimensions of the cuboid =

(ii) The total surface area of the cubical box =

The total surface area of the cuboidal box =

Clearly, the total surface area of a cuboidal box is greater than the cubical box.

Difference between them =

Given, dimensions of the greenhouse =

Area of the glass =

Therefore, the area of glass is

Given, dimensions of the greenhouse =

(ii) Tape needed for all the 12 edges = Perimeter =

Therefore,  of tape is needed for the edges.

Given,

Dimensions of the bigger box = ,

Dimensions of smaller box =

We know,

Total surface area of a cuboid =

Total surface area of the bigger box =

Area of the overlap for the bigger box =

Similarly,

Total surface area of the smaller box =

Area of the overlap for the smaller box =

Since, 250 of each box is required,

Total area of carboard required =

Cost of  of the cardboard = Rs 4

Cost of  of the cardboard =

Therefore, the cost of the cardboard sheet required for 250 such boxes of each kind is

Given, Dimensions of the tarpaulin =

The required amount of tarpaulin = Lateral surface area of the shelter + Area of top

Required

Therefore,  tarpaulin is required.

## Solutions of NCERT for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.2

Given,

The curved surface area of the cylinder =

And, the height of the cylinder,

We know, Curved surface area of a right circular cylinder =

Therefore, the diameter of the cylinder =

Given,

Height of the cylindrical tank =

Base diameter =

We know,

The total surface area of a cylindrical tank =

Therefore, square metres of the sheet is Note: There are two surfaces, inner and outer.

Given,

Height of the cylinder,

Outer diameter  =

Inner diameter  =

Inner curved surface area =

Therefore, the inner curved surface area of the cylindrical pipe is Note: There are two surfaces, inner and outer.

Given,

Height of the cylinder,

Outer diameter  =

Inner diameter  =

Outer curved surface area =

Therefore, the outer curved surface area of the cylindrical pipe is Note: There are two surfaces, inner and outer.

Given,

Height of the cylinder,

Outer diameter  =

Inner diameter  =

Outer curved surface area =

Inner curved surface area =

Area of the circular rings on top and bottom =

The total surface area of the pipe =

Therefore, the total surface area of the cylindrical pipe is

Given,

The diameter of the cylindrical roller =

Length of the cylindrical roller =

The curved surface area of the roller =

Area of the playground =

Therefore, the required area of the playground =

Given,

Radius of the cylindrical pillar, r =

Height of the cylinder, h  =

We know,

Curved surface area of a cylinder =

Curved surface area of the pillar =

Now,

Cost of painting  of the pillar =

Cost of painting the curved surface area of the pillar =

Therefore, the cost of painting curved surface area of the pillar is

Given, a right circular cylinder

Curved surface area of the cylinder =

The radius of the base =

Let the height of the cylinder be

We know,

Curved surface area of a cylinder of radius  and height   =

Therefore, the required height of the cylinder is

Given,

The inner diameter of the circular well =

Depth of the well =

We know,

The curved surface area of a cylinder =

The curved surface area of the well =

Therefore, the inner curved surface area of the circular well is

Given,

The inner diameter of the circular well =

Depth of the well =

The inner curved surface area of the circular well is

Now, the cost of plastering the curved surface per  = Rs. 40

Cost of plastering the curved surface of   =

Therefore, the cost of plastering the well is

Given,

Length of the cylindrical pipe =

Diameter =

The total radiating surface will be the curved surface of this pipe.

We know,

The curved surface area of a cylindrical pipe of radius  and length  =

CSA of this pipe =

Therefore, the total radiating surface of the system is

Given, a closed cylindrical petrol tank.

The diameter of the tank =

Height of the tank =

We know,

The lateral surface area of a cylinder of radius  and height  =

The lateral surface area of a cylindrical tank =

Therefore, the lateral or curved surface area of a closed cylindrical petrol storage tank is

Given, a closed cylindrical petrol tank.

The diameter of the tank =

Height of the tank =

Now, Total surface area of the tank =

Now, let  of steel sheet be actually used in making the tank

Since  of steel was wasted, the left  of the total steel sheet was used to made the tank.
The total surface area of the tank =

Therefore,  of steel was actually used in making the tank. The diameter of the base =

Height of the cylinder  =

We know,

Curved surface area of a cylinder of radius  and height   =

Now, Cloth required for covering the lampshade = Curved surface area of the cylinder

Therefore,  cloth will be required for covering the lampshade.

Given, a cylinder with a base.

The radius of the cylinder =

Height of the cylinder =

We know,

The lateral surface area of a cylinder of radius  and height   =

Area of the cylindrical penholder = Lateral areal + Base area

Area of 35 penholders  =

Therefore, the area of carboard required is

## NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.3

Given,

Base diameter of the cone =

Slant height =

We know, Curved surface area of a cone

Required curved surface area of the cone=

Given,

Base diameter of the cone =

Slant height =

We know, Total surface area of a cone = Curved surface area + Base area

Required total surface area of the cone=

Given,

The curved surface area of a cone =

Slant height

(i) Let the radius of cone be

We know, the curved surface area of a cone=

Therefore, the radius of the cone is

Given,

The curved surface area of a cone =

Slant height

The radius of the cone is

(ii) We know, Total surface area of a cone = Curved surface area + Base area

Therefore, the total surface area of the cone is

Given,

Base radius of the conical tent  =

Height of the conical tent =

Slant height =

Therefore, the slant height of the conical tent is

Given,

Base radius of the conical tent  =

Height of the conical tent =

Slant height =

We know, Curved surface area of a cone

Curved surface area of the tent

Cost of  of canvas =

Cost of  of canvas =

Therefore, required cost of canvas to make tent is

Given,

Base radius of the conical tent =

Height of the tent =

We know,

Curved surface area of a cone =

Area of tarpaulin required = Curved surface area of the tent

Now, let the length of the tarpaulin sheet be

Since  is wasted, effective length =

Therefore, the length of the required tarpaulin sheet will be 63 m.

Given, a conical tomb

The base diameter of the cone =

Slant height

We know, Curved surface area of a cone

Now, Cost of whitewashing per  =

Cost of whitewashing per  =

Therefore, the cost of white-washing its curved surface of the tomb is  .

Given, a right circular cone cap (which means no base)

Base radius of the cone =

Height

We know, Curved surface area of a right circular cone

The curved surface area of a cap =

The curved surface area of 10 caps =

Therefore, the area of the sheet required for 10 caps =

Given, hollow cone.

The base diameter of the cone =

Height of the cone =

Slant height =

We know, Curved surface area of a cone =

The curved surface area of 1 cone =

The curved surface area of 50 cones

Now, the cost of painting  area =

Cost of the painting  area

Therefore, the cost of painting 50 such hollow cones is

## Solutions of NCERT for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.4

We know,

The surface area of a sphere of radius  =

Required surface area =

We know,

The surface area of a sphere of radius  =

Required surface area =

We know,

The surface area of a sphere of radius  =

Required surface area =

Given,

The diameter of the sphere =

We know,

The surface area of a sphere of radius  =

Required surface area =

Given,

The diameter of the sphere =

We know,

The surface area of a sphere of radius  =

Required surface area =

Given,

The diameter of the sphere =

We know,

The surface area of a sphere of radius  =

Required surface area =

We know,

The total surface area of a hemisphere = Curved surface area of hemisphere + Area of the circular end

The required total surface area of the hemisphere =

Given,

We know,

The surface area of a sphere of radius  =

The ratio of surface areas of the ball in the two cases =

Therefore, the required ratio is

Given,

The inner radius of the hemispherical bowl =

We know,

The curved surface area of a hemisphere =

The surface area of the hemispherical bowl =

Now,

Cost of tin-plating  =  Rs 16

Cost of tin-plating  =

Therefore, the cost of tin-plating it on the inside is

Given,

The surface area of the sphere =

We know,

The surface area of a sphere of radius  =

Therefore, the radius of the sphere is

Let diameter of Moon be  and diameter of Earth be

We know,

The surface area of a sphere of radius  =

The ratio of their surface areas =

Therefore, the ratio of the surface areas of the moon and earth is

Given,

The inner radius of the bowl =

The thickness of the bowl =

We know, Curved surface area of a hemisphere of radius

The outer curved surface area of the bowl =

Therefore, the outer curved surface area of the bowl is Given,

The radius of the sphere =

Surface area of the sphere = Given,

The radius of the sphere =

The surface area of the sphere =

According to the question, the cylinder encloses the sphere.

Hence, the diameter of the sphere is the diameter of the cylinder.

Also, the height of the cylinder is equal to the diameter of the sphere.

We know, the curved surface area of a cylinder =

Therefore, the curved surface area of the cylinder is The surface area of the sphere =

And,  Surface area of the cylinder =

So, the ratio of the areas =

## NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.5

Given,

Dimensions of a matchbox =

We know,

The volume of a cuboid =

The volume of a matchbox=

Volume of 12 such matchboxes =

Therefore, the volume of a packet containing 12 matchboxes is

Given,

Dimensions of the cuboidal water tank =

We know,

Volume of a cuboid =

Volume of the water tank =

We know,

Volume of water the tank can hold =

Let the height of the vessel be

Dimensions of the cuboidal water tank =

We know,

The volume of a cuboid =

The volume of the water tank =

According to question,

Therefore, the required height of the vessel is

Given,

Dimensions of the cuboidal pit =

We know , Volume of a cuboid =

Volume of the cuboidal pit =

Now, Cost of digging  = Rs. 30

Cost of digging  =

## Q5 The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively and 10 m.

Given,

Length of the tank =

Depth of the tank =

Let the breadth of the tank be

We know, Volume of a cuboid =

The volume of the cuboidal tank =

We know,

Therefore, the breadth of the tank is

Given,

Dimensions of the tank =

4000 people requiring 150 litres of water per head per day

The total volume of water required =

Let the number of days the water will last be

We know,

The volume of a cuboid =

The volume of the water tank =

According to question,

Therefore, the water in the tank will last for 3 days.

Given,

Dimensions of the godown =

Dimension of each wooden crate =

We know , Volume of a cuboid =

Volume of the godown =

Volume of the each crate=

Let number of wooden crates be

Volume of  wooden crates = Volume of the godown

This is an important question.

Given,

Side of a solid cube =

We know, the volume of a cube of side  =

The volume of the given cube =

Now, the cube is cut into 8 equal cubes of side  (let)

The total volume of these 8 cubes = Volume of the bigger cube

Therefore, the side of the new cube is

Now, we know,

The surface area of a cube of side  =

The ratio between their surface areas

Therefore, the ratio of their surface areas is

Given,
Rate of water flow =

Depth of river,
Width of the river,
The volume of water flowing in 1 min =

Therefore, water falling into the sea in a minute =

## Solutions of NCERT for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.6

Given,

Circumference of the base of a cylindrical vessel =

Height =

Let the radius of the base of the cylinder be

Now, Circumference of the circular base of the cylinder =

The volume of the cylinder =

Also,

Therefore, the cylindrical vessel can hold  of water.

Given, A hollow cylinder made of wood.

The inner diameter of the cylindrical wooden pipe =
Outer diameter =

Length of the pipe =

The volume of the wooden cylinder =

Mass of  wood =
Mass of  wood =

Therefore, the mass of the wooden hollow cylindrical pipe is

Given,

(i) The dimension of the rectangular base of the tin can =

Height of the can =

The volume of the tin can =

(ii) The radius of the circular base of the plastic cylinder =

Height of the cylinder =

The volume of the plastic cylinder =

Clearly, the plastic cylinder has more capacity than the rectangular tin can.

The difference in capacity =

Given,

The lateral surface area of the cylinder =

Height of the cylinder =

(i) Let the radius of the base be

We know,

The lateral surface area of a cylinder =

Therefore, the radius of the base is

Given,

The lateral surface area of the cylinder =

Height of the cylinder =

The radius of the base is

(ii) We know,

The volume of a cylinder =

Therefore, the volume of the cylinder is .

(i) Given,

Rs 20 is the cost of painting  area of the inner curved surface of the cylinder.

Rs 2200 is the cost of painting area of the inner curved surface of the cylinder.

The inner curved surface area of the cylindrical vessel =

The inner curved surface area of the cylindrical vessel =

Height of the cylinder =

Let the radius of the circular base be

The inner curved surface area of the cylindrical vessel =

Therefore, the radius of the base of the vessel is

(iii) Height of the cylinder =

Radius of the base of the vessel =

Volume of the cylindrical vessel =

Therefore, the capacity of the cylindrical vessel is

(Using capacity(volume), we will find the radius and then find the surface area)

The capacity of the vessel = Volume of the vessel = litres

Height of the cylindrical vessel =

Let the radius of the circular base be

The volume of the cylindrical vessel =

Therefore, the total surface area of the vessel =

Therefore, square metres of metal sheet needed to make it is

Given,

Length of the cylindrical pencil =

The radius of the graphite (Inner solid cylinder)  =

Radius of the pencil (Inner solid graphite cylinder + Hollow wooden cylinder) =

We know, Volume of a cylinder=

The volume of graphite =

And, Volume of wood =

Therefore, the volume of wood is  and the volume of graphite is

Given,

Height =

The radius of the cylindrical bowl  =

The volume of soup in a bowl for a single person =

The volume of soup given for 250 patients =

Therefore, the amount of soup the hospital has to prepare daily to serve 250 patients is

## NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.7

Given,

Height =

We know,

Volume of a right circular cone =

Required volume =

Given,

Height =

We know,

Volume of a right circular cone =

Required volume =

Given,

Slant height =

Height =

We know,
Volume of a right circular cone =

Volume of the vessel=

Required capacity of the vessel =

Given,

Height =

Slant height =

We know,
Volume of a right circular cone =

Volume of the vessel=

Required capacity of the vessel =

Given,

Height of the cone =

Let the radius of the base of the cone be

We know,
The volume of a right circular cone =

Given,

Height of the cone =

Let the radius of the base of the cone be

We know,
The volume of a right circular cone =

Therefore the diameter of the right circular cone is

Given,

Depth of the conical pit  =

The top radius of the conical pit =

We know,
The volume of a right circular cone =

The volume of the conical pit =

Now,

The capacity of the pit =

Given, a right circular cone.

The radius of the base of the cone =

The volume of the cone =

(i) Let the height of the cone be

We know,
The volume of a right circular cone =

Therefore, the height of the cone is

Given, a right circular cone.

The volume of the cone =

The radius of the base of the cone =

And the height of the cone =

(ii) We know, Slant height,

Therefore, the slant height of the cone is .

Given, a right circular cone.

The radius of the base of the cone =

And Slant height of the cone =

(iii) We know,

The curved surface area of a cone =

Required curved surface area=

When a right-angled triangle is revolved about the perpendicular side, a cone is formed whose,

Height of the cone = Length of the axis=

Base radius of the cone =

And, Slant height of the cone =

We know,

The volume of a cone =

The required volume of the cone formed =

Therefore, the volume of the solid cone obtained is

## Q8 If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.

When a right-angled triangle is revolved about the perpendicular side, a cone is formed whose,

Height of the cone = Length of the axis=

Base radius of the cone =

And, Slant height of the cone =

We know,

The volume of a cone =

The required volume of the cone formed =

Now, Ratio of the volumes of the two solids =

Therefore, the required ratio is

Given,

Height of the conical heap =

Base radius of the cone =

We know,

The volume of a cone =

The required volume of the cone formed =

Now,

The slant height of the cone =

We know, the curved surface area of a cone =

The required area of the canvas to cover the heap  =

Solutions of NCERT for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.8

Given,

The radius of the sphere =

We know, Volume of a sphere =

The required volume of the sphere =

Given,

The radius of the sphere =

We know, Volume of a sphere =

The required volume of the sphere =

## Q2 (i) Find the amount of water displaced by a solid spherical ball of diameter 28 cm

The solid spherical ball will displace water equal to its volume.

Given,

The radius of the sphere =

We know, Volume of a sphere =

The required volume of the sphere =

Therefore, the amount of water displaced will be

The solid spherical ball will displace water equal to its volume.

Given,

The radius of the sphere =

We know, Volume of a sphere =

The required volume of the sphere =

Therefore, amount of water displaced will be

Given,

The radius of the metallic sphere =

We know, Volume of a sphere =

The required volume of the sphere =

Now, the density of the metal is  per ,which means,

Mass of  of the metallic sphere =

Mass of  of the metallic sphere =

Given,

Let  be the diameters of Earth

The diameter of the Moon =

We know, Volume of a sphere =

The ratio of the volumes =

Therefore, the required ratio of the volume of the moon to the  volume of the earth is

The radius of the hemispherical bowl =

We know, Volume of a hemisphere =

The volume of the given hemispherical bowl =

The capacity of the hemispherical bowl =

Given,

Inner radius of the hemispherical tank =

Thickness of the tank =

We know, Volume of a hemisphere =

Volume of the iron used = Outer volume - Inner volume

Given,

The surface area of the sphere =

We know, Surface area of a sphere =

The volume of the sphere =

Given,

is the cost of white-washing  of the inside area

is the cost of white-washing  of inside area

(i) Therefore, the surface area of the inside of the dome is

Let the radius of the hemisphere be

Inside the surface area of the dome =

We know, Surface area of a hemisphere =

The volume of the hemisphere =

Given,

The radius of a small sphere =

The radius of the bigger sphere =

The volume of each small sphere=

And, Volume of the big sphere of radius

According to question,

Given,

The radius of a small sphere =

The surface area of a small sphere =

The radius of the bigger sphere =

The surface area of the bigger sphere =

And,

We know, the surface area of a sphere =

The ratio of their surface areas =

Therefore, the required ratio is

Given,

The radius of the spherical capsule =

The volume of the capsule =

Therefore,  of medicine is needed to fill the capsule.

## NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Excercise: 13.9 External dimension od the bookshelf =

(Note: There is no front face)

The external surface area of the shelf =

We know, each stripe on the front surface is also to be polished. which is 5 cm stretch.

Area of front face =

Area to be polished =

Cost of polishing  area =

Cost of polishing  area =

Now,

Dimension of inner part =

Area to be painted in 3 rows =

Cost of painting  area =

Cost of painting  area =

Total expense required for polishing and painting = Given,

The radius of the wooden spheres =

The surface area of a single sphere =

Again, the Radius of the cylinder support =

Height of the support =

The base area of the cylinder =

Now, Cost of painting silver =

Cost of painting 1 wooden sphere = Cost of painting silver

Now, Curved surface area of the cylindrical support =

Now, Cost of painting  black =

Cost of painting 1 such stand = Cost of painting silver =

The total cost of painting 1 sphere and its support =

Therefore, total cost of painting 8 such spheres and their supports =

Let the radius of the sphere be

Diameter of the sphere =

According to question,

Diameter is decreased by

So, the new diameter =

New surface area =

Decrease in surface area =

Percentage decrease in the surface area =

## NCERT solutions for class 9 maths chapter wise

 Chapter No. Chapter Name Chapter 1 NCERT solutions for class 9 maths chapter 1 Number Systems Chapter 2 CBSE NCERT solutions for class 9 maths chapter 2 Polynomials Chapter 3 Solutions of NCERT class 9 maths chapter 3 Coordinate Geometry Chapter 4 NCERT solutions for class 9 maths chapter 4 Linear Equations In Two Variables Chapter 5 CBSE NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry Chapter 6 Solutions of NCERT class 9 maths chapter 6 Lines And Angles Chapter 7 NCERT solutions for class 9 maths chapter 7 Triangles Chapter 8 CBSE NCERT solutions for class 9 maths chapter 8 Quadrilaterals Chapter 9 Solutions of NCERT class 9 maths chapter 9 Areas of Parallelograms and Triangles Chapter 10 NCERT solutions for class 9 maths chapter 10 Circles Chapter 11 CBSE NCERT solutions for class 9 maths chapter 11 Constructions Chapter 12 Solutions of NCERT class 9 maths chapter 12 Heron’s Formula Chapter 13 NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Chapter 14 CBSE NCERT solutions for class 9 maths chapter 14 Statistics Chapter 15 Solutions of NCERT class 9 maths chapter 15 Probability

## How to use NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes?

• Revise the formulas and concepts regarding the figures circle, rectangle, square, etc.

• Learn about the other figures introduced in this chapter.

• Memorize the formulas for every shape.

• Learn the formula application by going through some examples.

• Jump on to the practice exercises to get command over the topic.

• You can use NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes during the practice.

Keep working hard and happy learning!